A Fully Analog Implementation of Model Predictive Control with Application to Buck Converters

arXiv:2509.05463 · eess.SY, cs.SY · Submitted 2025-09-05 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa, and with me are Dev and Taro, guest researcher.

Dev: Today's paper: "A Fully Analog Implementation of Model Predictive Control with Application to Buck Converters".

Rosa: The gist This paper proposes a novel approach to design analog electronic circuits that implement Model Predictive Control (MPC) policies for dynamical systems described by affine models <ref:2509.05463#pg2> and demonstrates…

Dev: First, who's behind it and why it matters.

Title and authors: Rosa: We’ve looked at the paper "A Fully Analog Implementation of Model Predictive Control with Application to Buck Converters" and seen how they tackle building MPC policies in analog circuits using techniques like explicit MPC, and the complexity reduction strategies they employ.

Dev: It’s clear that the authors are trying to bridge the gap between theoretical control design and practical, low-latency analog implementation for power electronics.

Taro: I wonder if this approach is just a clever way to avoid digital hardware, or if it fundamentally changes how we think about implementing complex controllers in embedded systems.

Rosa: It seems like they are using established MPC formulations but forcing them into a structure that can be mapped onto discrete analog operations, which is what makes the implementation feasible.

Dev: The focus on the explicit MPC closed-form policy represented as a continuous piecewise affine function pi(p k) over regions R i is where they start simplifying the problem before moving to hardware design.

Taro: So, when you look at those region reduction techniques—like move blocking or optimal merging of regions—are those just mathematical tricks, or do they actually translate into real physical simplifications in the circuit?

Rosa: They are designed to minimize the number of decision variables and facets defining those regions, which directly reduces the complexity of the control law that needs to be implemented in silicon.

Dev: That reduction is what allows them to move from a potentially intractable problem for digital solvers into something manageable for analog circuitry.

Taro: What about when you consider the application itself, like modeling DC-DC Buck converters as a cascade of subsystems S1 and S2? Is that just a standard way to look at it?

Rosa: For this paper, yes, they model the system by discretizing and linearizing it under Assumption three to get that DT affine system structure they need for the MPC formulation <ref:2509.05463#pg3>.

Dev: That linearization step is crucial because MPC requires an affine model to work efficiently; without that assumption, you’re back to a much more complicated non-linear problem.

Taro: So, what’s the main takeaway for someone who just reads this summary? Is it just "it works," or is there a deeper insight into how these systems interact?

Rosa: It's more than that; it shows that you can achieve robust performance in real power conversion applications by marrying advanced control theory with practical, low-cost analog circuitry.

Dev: It’s about taking something theoretically flexible like MPC and making it physically realizable in a way that is fast and relatively simple to build compared to high-end digital solutions.

The paper's summary: Rosa: We discussed the setup, but let’s get into what the actual summary of "A Fully Analog Implementation of Model Predictive Control with Application to Buck Converters" says about how it works operationally.

Dev: The core idea is that they formulate the optimal control problem as minimizing a cost function (2a) that includes regulation terms and a penalty on input variation, which then leads to a compact quadratic program <ref:2509.05463#pg3>.

Taro: So, the mathematical formulation itself is standard MPC stuff, but the authors are really focused on how they transform that abstract optimization into something that can be calculated in real-time by hardware.

Rosa: Exactly; they transform it into a one-step receding horizon policy where the optimal input u* k is given by pi(p k), which is this continuous piecewise affine function defined over regions R i.

Dev: That means at any time instant k, you just look up which region the current parameter vector falls into and use the corresponding simple affine control law.

Taro: If you’re driving, what does that mean for me? Does it mean my car is constantly recalculating its optimal control input based on where it is in the state space?

Rosa: Not exactly constant recalculation; it means at each step, you evaluate the policy function pi(p k) using those pre-calculated regions, and that’s what feeds the analog circuit.

Dev: And they apply this to Buck converters by modeling them as two subsystems and then defining constraints on the duty cycle d within the range of zero percent to one hundred percent <ref:2509.05463#pg2>.

Taro: That constraint on the duty cycle is a big deal because it shows they aren't just doing abstract math; they are solving a tangible problem in power conversion where you can’t just set anything arbitrarily.

Rosa: Right, and for the disturbances, they use an algebraic estimator to estimate the output current disturbance nu one(t) using something based on linear estimation <ref:2509.05463#pg1>.

Dev: So, they have a mechanism to get an estimate of what’s happening outside the loop so the MPC can react proactively rather than just reacting after things have already changed.

The paper's improvements: Taro: We’ve looked at the summary and now we need to talk about what improvements this methodology suggests for future work or for implementing it better.

Rosa: The paper explicitly focuses on applying specific complexity reduction strategies to minimize the circuit cost, specifically mentioning the move blocking strategy and optimal merging of regions.

Dev: Those are techniques designed to tighten the upper bound on the number of regions R, which is a direct way to reduce hardware complexity by shrinking those decision spaces.

Taro: And they also mention hyperplane separation for saturated regions, which is another way to trim down the policy definition without losing too much control authority.

Rosa: And then there's removing trivial inequalities; they noted that some inequalities associated with the facets of the parameter domain P are always satisfied and can be eliminated.

Dev: That’s a clean way to simplify the mathematical representation before it gets translated into circuit logic, ensuring you’re only building what actually matters for the physical system.

Taro: So, these aren't just suggestions for better math; they are concrete steps to make the final analog implementation more efficient and cheaper.

Rosa: Exactly; they aim to reduce complexity by removing unnecessary constraints from the model before it gets mapped onto the actual circuit components, which is a very practical improvement.

Conclusion: Dev: So, wrapping up on "A Fully Analog Implementation of Model Predictive Control with Application to Buck Converters," this paper proposes a general methodology for implementing explicit MPC policies as fully analog electronic circuits.

Rosa: It achieves effective rejection of line and load disturbances while maintaining fast sampling rates and avoiding the digital overhead that we talked about earlier.

Taro: I just want to add that the main thing here is confirming the feasibility of getting this complex control structure running in a high-frequency environment using only analog hardware.

Dev: It does, it validates this approach through simulations showing excellent disturbance rejection performance across a wide range of uncertainties, and confirms it’s feasible for high-frequency operation.

Rosa: So, the conclusion is that we can implement these advanced MPC policies in a low-cost way using components like comparators and operational amplifiers, which is a big step for power control design.

Politecnico di Torino Department of Computer and Control Engineering · Politecnico di Torino Department of Electronics and Telecommunications · King Abdullah University of Science and Technology

eess.SY, cs.SY

Submitted: 2025-09-05

Updated: 2026-10-15

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 72/100

The gist: The gist This paper proposes a novel approach to design analog electronic circuits that implement Model Predictive Control (MPC) policies for dynamical systems described by affine models

Key concepts

Model Predictive Control (MPC): MPC is a control strategy that uses a mathematical model of a system to predict future behavior. It calculates the best sequence of control actions over a short future time horizon by minimizing an objective function, allowing the system to proactively manage disturbances.
Explicit MPC (EMPC): EMPC converts the complex optimization problem from MPC into a fixed, continuous mathematical formula. This formula is represented as a piecewise affine function, which simplifies the control policy into distinct regions. This structure is crucial for designing efficient analog circuits.
Move Blocking Strategy: This strategy reduces the number of variables needed in the MPC formulation by grouping similar inputs. By doing this, it significantly lowers the total number of decision variables required to define the control regions, which directly simplifies and speeds up the resulting analog circuit design.
DC-DC Buck Converters: These are electronic circuits used to efficiently convert a higher DC voltage into a lower, regulated DC voltage. The paper applies its MPC controller to manage these converters, ensuring stable and precise output voltage regulation while handling external disturbances like line noise.

Terminology

Model Predictive Control — 2× in this paper
Model Predictive Control (MPC) — 1× in this paper · explained in 7 episodes

Summary

The gist This paper proposes a novel approach to design analog electronic circuits that implement Model Predictive Control (MPC) policies for dynamical systems described by affine models <ref:2509.05463#pg2> and demonstrates its effectiveness through application to DC-DC Buck converters <ref:2509.05463#pg3>.

Model Predictive Control Formulation

The MPC problem is formulated for a discrete-time (DT) affine system described by equations (1a) and (1b), which define the state, input, and output vectors <ref:2509.05463#pg4>. The optimal control problem involves minimizing a cost function (2a) composed of regulation terms and a penalty on input variation <ref:2509.05463#pg5>. This problem is transformed into a compact QP form (94a) comprising only the input sequence as decision variables <ref:2509.05463#pg9>. The policy is defined by the one-step receding horizon policy, where the optimal input u∗k is given by π(pk) <ref:2509.05463#pg10>.

Explicit MPC Design and Complexity Reduction

The paper leverages the explicit MPC (EMPC) closed-form policy, which is represented as a continuous piecewise affine (PWA) function π(pk) defined over regions Ri <ref:2509.05463#pg8>. To minimize circuit cost, several complexity-reduction strategies are employed:

Move Blocking Strategy [31] reduces the number of decision variables from Npnu to Ncnu, yielding a tighter upper bound on the number of regions R.
Optimal Merging of Regions merges regions sharing the same affine control law, leading to fewer facets defining the region <ref:2509.05463#pg4>.
Hyperplane Separation of Saturated Regions defines a separation function σ(p) = a⊤p+b to reduce complexity by removing saturated regions.
Removal of Trivial Inequalities eliminates inequalities associated with the facets of the parameter domain P, which are always satisfied <ref:2509.05463#pg8>.

Analog Circuit Implementation

The reduced-complexity EMPC policy is implemented using commercially available low-latency analog components. The circuit architecture consists of a one multiplexer (MUX), generalized adders, comparators, and a small logic gate network.

The MUX selects the control input based on the computed Boolean signals rj and s.
Generalized adders generate the affine functions u(i) = Kip + li, where p is the vector of EMPC parameters.
Comparators implement the Boolean signals rj and s by evaluating inequalities of the form [α1,..., αnp+1]p + αnp+1V0 ≥ 0.

The overall circuit achieves a total latency of approximately 1 µs, which is shorter than the switching period T = 2 µs for high-frequency operation.

Application to DC-DC Buck Converters

The approach is applied to the control of DC-DC Buck converters, modeling them as a cascade of two subsystems (S1 and S2). The system is discretized and linearized under Assumption 3, leading to a DT affine system.

Disturbances estimation involves estimating the output current disturbance ν1(t) = io(t) using a linear estimator based on algebraic design. The state estimation for the capacitor voltage vC(t) is achieved via a simple algebraic estimator, yielding ˆvC = RCo(ˆio − iL) + RCoRL+1/vo.
The final controller input is constructed as uk = π([iL,k, vˆC,k, ˆio,k, vin,k]⊤), imposing constraints on the duty cycle d in the range [0%, 100%].

Robust Performance Assessment

The controller's local stability is guaranteed by Theorem 3, which establishes a condition based on the matrix P and disturbance bounds. Extensive Monte Carlo simulations show that the EMPC policy achieves consistently good performance across component uncertainties. Specifically, the system exhibits outstanding disturbance-rejection performance, outperforming standard methods <ref:2509.05463#pg2>. Circuit-level simulations confirm that the impact of circuit non-idealities is negligible compared to high-level simulations.

Conclusion

This paper introduces a general methodology for implementing explicit model predictive control policies as fully analog electronic circuits <ref:2509.05463#pg9>. The proposed approach achieves effective rejection of load and line disturbances while maintaining fast sampling rates and avoiding digital overhead <ref:2509.05463#pg2>. The EMPC analog circuit is validated through simulations, demonstrating excellent disturbance rejection performance across a wide range of uncertainties. The paper concludes by confirming the feasibility of the proposed approach for high-frequency operation <ref:2509.05463#pg9>.

Improvements for AI systems

Bold header: Analog MPC Policy Implementation

This allows AI systems to implement Model Predictive Control (MPC) policies using only a limited set of low-latency, commercially available components, circumventing the significant overhead in terms of costs and circuit area due to the need to employ expensive digital hardware.

Bold header: High-Frequency Real-Time Control

The system can achieve a high-frequency sampling at 500 kHz, which is compatible with modern Buck designs and considerably improves upon previous solutions reported in [23, 24, 25, 26], enabling control over fast dynamical systems.

Bold header: Robust Disturbance Rejection

The controller can achieve outstanding disturbance-rejection performance, demonstrating the ability to handle unpredictable and sudden variations of the input supply voltage (line disturbances) and of the current drawn by the load (load disturbances).

Bold header: Parametric Uncertainty Handling

The system can maintain consistently good performance across the entire range of admissible component values when subjected to parametric uncertainty in passive components like RL, L, Co, and RCo.

Bold header: Low-Latency Control Loop

The analog circuit exhibits a total propagation delay of 1 µs, which is shorter than the switching period T = 2 µs (fsw = 500 kHz), confirming feasibility for high-frequency operation and significantly improving upon previous solutions reported in [12].

Bold header: Optimized Complexity Reduction

The methodology can be tailored to minimize circuit complexity by applying strategies such as the move blocking strategy, the non-disjoint optimal merging of regions, and hyperplane separation of saturated regions, resulting in an 89% reduction in the number of regions for the control policy.

Bold header: Low-Cost, Hardware-Efficient Design

The implementation relies on components like comparators, resistors, operational amplifiers (OP-AMPs), and one multiplexer (MUX), enabling a low-cost circuit design compared to approaches that require custom hardware or complex digital solutions.

Abstract

This paper proposes a novel approach to design analog electronic circuits that implement Model Predictive Control (MPC) policies for dynamical systems described by affine models. Effective approaches to define a reduced-complexity explicit MPC are combined and applied to realize an analog circuit comprising a limited set of low-latency, commercially available components. The practical feasibility and effectiveness of the proposed approach are demonstrated through its application in the design of a novel controller for DC-DC Buck converters. We formally analyze the stability of the resulting system and conduct extensive numerical simulations to demonstrate the control system's performance in rejecting line and load disturbances.