Notethattheaccuracyof δl 1 isnotmeasureddirectlyforReLU because it does not have an explicit inversion. This precludes stability forα = 0 and dl > dl 1 (expanding layer), as the matrix productWlBl will be singular. Forexample,inthenonlinearregression experiment shown in the main text, we initialize the SLDI feedback asB = B1B2, whereB1 and B2 are the feedback matrices for sequential DI. Once again, when the controller has no leak, this will produce the same steady state assequential dynamicinversion. We study a simple case here as an illustration, and leave a more thorough analysis for futurework.

Learning depends on changes in synaptic connections deep inside the brain. In multilayer networks, these changes are triggered by error signals fed back from the output, generally through a stepwise inversion of the feedforward processing steps. The gold standard for this process --- backpropagation --- works well in artificial neural networks, but is biologically implausible. Several recent proposals have emerged to address this problem, but many of these biologically-plausible schemes are based on learning an independent set of feedback connections. This complicates the assignment of errors to each synapse by making it dependent upon a second learning problem, and by fitting inversions rather than guaranteeing them.

Learning depends on changes in synaptic connections deep inside the brain.

Accurate convergence of the backward pass dynamics is crucial for the success of dynamic inversion. As noted in Section 3.2, the eigenvalues of the matrix Here we provide more details for single-loop dynamic inversion (SLDI). As mentioned in the main text, we suspect that dynamic inversion may relate to second-order methods. Following the derivations in Botev et al. (2017), we can write the block-diagonal sample A more thorough analysis is merited on the relationship between Eqs. This was not necessary for MNIST classification or MNIST autoencoding.

Learning depends on changes in synaptic connections deep inside the brain.

As the authors note, the stability of the feedback dynamics depends on a condition on the eigenvalues of WB - alpha*I. Without it, the feedback dynamics will yield unpredictable results and presumably not perform effective credit assignment. This condition is extremely unlikely to be satisfied generically, and is essentially the analog of sign-symmetry in forward and backward weights when one considers pseudoinverses rather than transposes. The authors manually enforce that it be satisfied at initialization, and manually adjust the backward weights if the condition is violated during training. These manual initialization choices and adjustments are doing much of the work of credit assignment in the authors' algorithm -- I can't tell from the results as presented how helpful the dynamic inversion really is.

The reviewers all agreed that this is a worthwhile contribution to the biologically realistic credit assignment problem and a consensus on "Accept" was achieved easily.

Learning depends on changes in synaptic connections deep inside the brain. In multilayer networks, these changes are triggered by error signals fed back from the output, generally through a stepwise inversion of the feedforward processing steps. The gold standard for this process --- backpropagation --- works well in artificial neural networks, but is biologically implausible. Several recent proposals have emerged to address this problem, but many of these biologically-plausible schemes are based on learning an independent set of feedback connections. This complicates the assignment of errors to each synapse by making it dependent upon a second learning problem, and by fitting inversions rather than guaranteeing them.

Improving robustness to uncertainty and rejection of external disturbances represents a significant challenge in aerial robotics. Nonlinear controllers based on Incremental Nonlinear Dynamic Inversion (INDI), known for their ability in estimating disturbances through measured-filtered data, have been notably used in such applications. Typically, these controllers comprise two cascaded loops: an inner loop employing nonlinear dynamic inversion and an outer loop generating the virtual control inputs via linear controllers. In this paper, a novel methodology is introduced, that combines the advantages of INDI with the robustness of linear structured $\mathcal{H}_\infty$ controllers. A full cascaded architecture is proposed to control the dynamics of a multirotor drone, covering both stabilization and guidance. In particular, low-order $\mathcal{H}_\infty$ controllers are designed for the outer loop by properly structuring the problem and solving it through non-smooth optimization. A comparative analysis is conducted between an existing INDI/PD approach and the proposed INDI/$\mathcal{H}_\infty$ strategy, showing a notable enhancement in the rejection of external disturbances. It is carried out first using MATLAB simulations involving a nonlinear model of a Parrot Bebop quadcopter drone, and then experimentally using a customized quadcopter built by the ENAC team. The results show an improvement of more than 50\% in the rejection of disturbances such as gusts.