Weaknesses
W1) The introduction makes some imprecise or misleading statements about the state of the art as well as technical aspects. All statements should be carefully checked and (where needed) be backed up with proper argumentation or references. Examples:
* ll. 33-34: "Currently, learning-based approaches mostly parameterize nonlinear controllers as neural networks and train them to optimize performance". While NN are a popular function approximator in learning-based control, they are actually used in different flavors (model learning, controller parametrization, surrogate model, etc.). Furthermore, there are many other ML techniques that are also used, such as Gaussian processes, which are particularly popular when it comes to guarantees (examples: A Lederer, J Umlauft, S Hirche, "Uniform error bounds for Gaussian process regression with application to safe control"; C Fiedler, CW Scherer, S Trimpe, "Learning-enhanced robust controller synthesis with rigorous statistical and control-theoretic guarantees") Hence, the characterization of the area of learning-based control as described in this above sentence is too narrow in my opinion.
* ll. 47-49: "...often overlooking the steady-state behavior (cost at the equilibrium state)." The statement is not supported by references. In fact, I would disagree with it, as there are works that specifically focus on the long term performance of controllers or RL policies (examples: A Doerr et al, "Optimizing long-term predictions for model-based policy search"; N Lambert et al, "Learning accurate long-term dynamics for model-based reinforcement learning")
* ll. 49-51: The description of classical P- and I-control is inaccurate. While it is correct that I-control is often used in practice to make steady-state error zero, the actual behavior depends on the controller *and* the dynamical process. Depending on the process dynamics (e.g., if it has an integrator itself), also a P-controller may lead to zero steady-state error. Also, the statement that P-control is for the transient, but disregarding steady-state performance is inaccurate. P-control also affects the steady-state in general. Again, the general description of how P- and I-control are often used in practice is not wrong, but the authors should more carefully word these sentences to avoid technical inaccuracies.
* The statement of the main research question that is addressed in this paper is too general to be representative of the paper's contribution: "Can we learn nonlinear controllers that guarantee transient stability and zero steady-state error for MIMO systems?" (ll. 55-56). First, the authors state themselves in the next sentence that this question can't be answered for general nonlinear system. Second, there are existing works in learning-based control that focus specifically on MIMO PI-control (see below). IMO, the research question should thus be rephrased to already clarify the scope of this work.
* ll. 67-68, "This class of functions is too restrictive to use for controller design" -- Why? Appropriate reasoning or reference is missing to support the statement.
W2) Related work: There is no explicit discussion of related work as a separate section/subsection, which is not a requirement of course, but could be helpful. Some references are mentioned in the introduction, but IMO the following aspects require further discussion:
* The control problem and the objectives that are stated are super general in control (stability, zero steady-state error, fast transient). Hence, there will be a lot of works in learning-based control addressing similar problems. While the paper does mention references and related work, I do not find this discussion very structured and have trouble pin-pointing what exactly is the difference to prior art and thus contribution of this paper. Just saying, for example, that "guaranteeing stability and output tracking" (l. 144) is not helpful in this regard as the statement is too general and not specific to the contribution at hand. It might help to first clearly specify the problem that is addressed, and afterwards discuss the related work specifically to the problem.
* Given that the consideration of MIMO systems is a main aspect of this work, there should be a review specific to this. Indeed, there are existing works in learning-based control that focus specifically on MIMO PI-control (e.g., A Doerr et al, "Model-based policy search for automatic tuning of multivariate PID controllers", I Carlucho, M De Paula, GG Acosta, "An adaptive deep reinforcement learning approach for MIMO PID control of mobile robots", and many other works on "self-tuning" for MIMO systems).
Overall, I do not get a clear idea how this work compares to the state of the art.
W3) Specification of controller structure: The control law u = g(y) is not precisely specified. Concretely, is g(.) a static function, or does it itself involve states and memory (i.e., is it itself dynamic)? This is critical, as dynamic controllers (such as combining a state estimator and state-feedback controller) are more powerful than static maps in general. Later on, it seems that static NN are used as components of g(.). However, the controller also involves an integrator, which would not be captured by a static function g(.) and is actually a dynamic component. If, on the other hand, dynamic output-feedback controllers are admissible, one could presumably learn much richer control behavior. IMO, it is ok to restrict the controller structure (from what I understand, to an integrator + static map), but it needs to be clearly and correctly stated.
W4) I am wondering whether it is really justified to call the proposed structure a proportional + integral (PI) controller. In particular, the term that is called "proportional" isn't really proportional to the error for a general nonlinear map. I can see that what the authors propose is a generalization of well-known PI-control structure. However, I find it a bit misleading to call this a "proportional control". In fact, p(-y+ybar) is a general nonlinear function of the tracking error. Hence, it will in general not be proportional to the tracking error. Please elaborate, why you call the architecture "PI", and ideally provide references if this is a common naming.
W5) I do not find the notion of "end-to-end" guarantees conclusive and well justified. The authors use this notion to denote the fact that they can provide certain guarantees a-priori before training (by combining a certain system class (passive) with the proposed controller architecture). As far as I know, the term "end-to-end" is not commonly used for this (if you have references where it is used in this way, please provide them). If a controller has theoretical guarantees, these are most of the time "by construction", so I don't understand why this is called "end-to-end".
* Minor: In the introduction, it is not yet clear what the authors mean when mentioning "end-to-end guarantees" on l. 80. It should be explained when introduced for the first time.
W6) The presentation of experiments in the main paper is not self-contained and, IMO, insufficient. There needs to be a basic description of the problem in the main paper, which can be understood without consulting the supplementary material. But this is currently missing. Because of this, one cannot judge how challenging the control problem actually is, and whether sophisticated nonlinear control is actually needed. Furthermore, the verification of assumptions is critical because a main claim of the paper are the theoretical guarantees. This should be in the main paper. As these are essential aspects of the contribution, the experimental section isn't self contained. If the authors cannot fit all essential components, then a conference might be the wrong format for this manuscript.
W7) Experimental results are not fully conclusive; they compare against a standard neural network, but more classical controller structures as baseline are missing in the main paper. Thus, I find the results only partially convincing for showing the benefit of the proposed Neural-PI controller.
* It remains unclear whether the chosen problems require sophisticated nonlinear control such as the proposed Neural-PI controller. First, the problem description does not convey this (see above). Second, in the supplementary material, the authors report the results also of a standard linear PI, whose parameters have been optimized similarly to the optimization of the NNs. Apparently, the linear PI achieves almost the same performance as the Neural-PI for vehicle platooning, and very close performance for the power grid. The results suggests that at least for these systems, there is no significant benefit of the proposed and more involved Neural-PI structure. Hence, the result are not conclusive regarding the proposed architecture in this regards.
* Baseline comparisons (i.e., the linear PI from the supplementary, and maybe also a model-based controller such as LQR) should be included in the main paper to judge the performance of the proposed Neural-PI.
* I find it surprising that the DenseNN cannot stabilize the vehicle platooning system and actually has some significant deviation at the end of the horizon (Fig. 5(b), 15 sec). Such deviation should also be visible in the objective function, so I would expect that the NN (if trained enough) can accommodate for this.
* A discussion of assumptions and theoretical guarantees is missing in the main paper. In ll. 177/178, it is claimed that in the experimental section, it will be discussed how EIP (the main assumption) can be verified if the system is unknown. While I agree that this discussion is very important with regards of the claimed contribution of this paper, I could not find this in Sec. 6.
* The above points hold for both examples (platooning, power grid).
Minor:
* l 192: As usually, p is a gain (e.g., a real number), it would be good to emphasize when introducing it that, here, it is meant as a function of the tracking error.