Minimum Attention MPC (MAMPC)

• MAMPC is a class of receding-horizon control formulations that jointly optimizes standard performance with the minimization of controller 'attention', such as update frequency or input variations.
• It employs diverse methodologies—including LP-based eCLF constraints, Brockett-style smoothness penalties, and RL-triggered policies—to effectively reduce computational and implementation burdens.
• Empirical studies reveal trade-offs between sparse control actions and tracking accuracy, driving ongoing research for robust, real-time, and resource-aware predictive control.

Searching arXiv for relevant papers on Minimum Attention Model Predictive Control and related minimum-attention control formulations. Minimum Attention Model Predictive Control (MAMPC) denotes a family of receding-horizon control formulations in which controller “attention” is optimized jointly with closed-loop performance and constraints. In the cited literature, attention is defined in several technically distinct ways: as the inverse of the time elapsed between controller executions, as the number of nonzero input changes over a horizon, as the frequency of MPC recomputation, or as the rate of variation of the control law with respect to state and time. Despite these differences, the common objective is to reduce implementation burden while preserving stabilization, tracking, economic performance, or constraint satisfaction in a model-based predictive framework (Donkers et al., 2011, Lee et al., 2019, Bøhn et al., 2020, Teja et al., 28 Jul 2025).

1. Origins and conceptual scope

Minimum-attention control predates the explicit MAMPC terminology. In the linear sampled-data setting, minimum attention control (MAC) and anytime attention control (AAC) were formulated for linear plants with scarce computational and communication resources, with “attention” interpreted as the inverse of the time elapsed between two consecutive executions of a control task. The corresponding control laws were posed as online linear programs built from infinity-norm-based extended control Lyapunov functions (eCLFs), with admissible inter-execution times restricted to a finite set $H=\{h_1,\dots,h_M\}$ (Donkers et al., 2011).

A second lineage treats attention as a functional of the control law itself. Brockett’s minimum attention functional penalizes the rate of change of the control with respect to both state and time. Under the structured law

$u(t,x)=v(t)+K(t)x,$

the corresponding terms become

$$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$

and the control design problem is posed on a finite horizon using a Liouville-equation representation of the nonlinear dynamics (Lee et al., 2019).

A third lineage emerges from predictive control architectures that optimize the MPC update interval rather than the plant input sequence alone. In the three-part architecture consisting of a high-level MPC plan generator, a low-level linear state-feedback compensator, and a recomputation policy learned by reinforcement learning, attention is the frequency or energetic cost of MPC recomputations. The update interval is not optimized as an internal MPC decision variable; instead, it is induced by a binary recomputation policy interacting with the plant and the planner (Bøhn et al., 2020).

The explicit term “Minimum Attention Model Predictive Control (MAMPC)” is used for a receding-horizon formulation in which the zero norm of the successive input differences is constrained and minimized together with standard stage costs for reference tracking. In that setting, MAMPC retains standard MPC prediction and constraint handling, but imposes sparsity directly on $\Delta u$ across an augmented horizon that includes past optimal inputs (Teja et al., 28 Jul 2025).

2. Meanings of “attention” and associated objectives

The central technical issue in MAMPC is that “attention” is not a single invariant quantity across the literature. The principal definitions are summarized below.

Formulation	Attention variable	Optimization role
MAC/AAC for linear systems	$a_k:=1/h_k$	Minimize attention by maximizing feasible hold $h$
Brockett-style minimum attention	$$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$	Penalize control-law variability over $\mathcal X\times[0,T]$
RL-triggered MPC update optimization	recomputation indicator $a\in\{0,1\}$	Penalize or energetically price MPC recomputation
Receding-horizon sparse-input-change MAMPC	$\|\Delta u\|_0$	Constrain the number of input changes over the horizon

In the LP-based minimum attention framework, the optimization variable is the next hold length $u(t,x)=v(t)+K(t)x,$0. MAC chooses $u(t,x)=v(t)+K(t)x,$1 and $u(t,x)=v(t)+K(t)x,$2 so as to minimize attention, equivalently maximize $u(t,x)=v(t)+K(t)x,$3, while guaranteeing a prescribed exponential decay rate. AAC fixes $u(t,x)=v(t)+K(t)x,$4 externally and optimizes performance under the available attention budget (Donkers et al., 2011).

In the Brockett-style formulation, attention is a Sobolev-type seminorm of the control mapping,

$u(t,x)=v(t)+K(t)x,$5

and the paper studies existence and computation of optimal laws within the class $u(t,x)=v(t)+K(t)x,$6. The finite-horizon objective augments this attention cost with a terminal density-tracking term $u(t,x)=v(t)+K(t)x,$7 (Lee et al., 2019).

In the RL-triggered MPC formulation, attention is the rate or cost of recomputing the high-level MPC plan. For the cart–pendulum, the stage reward is

$u(t,x)=v(t)+K(t)x,$8

so each recomputation $u(t,x)=v(t)+K(t)x,$9 incurs an explicit penalty. For the battery-storage system, recomputation energy enters the plant dynamics through the extra load $$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$0, with $$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$1, so attention reduces state of charge and therefore economic return (Bøhn et al., 2020).

In the receding-horizon sparse-input-change formulation, attention is the number of manipulated-input changes. The defining constraint is

$$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$2

where $$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$3 is the attention budget and $$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$4 is the sparsity horizon that includes past optimal inputs from previous horizons so that changes at the horizon boundary are counted consistently (Teja et al., 28 Jul 2025).

These definitions are related but not interchangeable. A plausible implication is that MAMPC should be understood as a design principle—minimization of implementation burden in a predictive architecture—rather than as a single canonical optimization problem.

3. Core formulations and control architectures

The LP-based minimum-attention formulation considers the continuous-time LTI plant

$$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$5

implemented with sample-and-hold control

$$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$6

where $$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$7. The eCLF is defined as

$$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$8

with $$\frac{\partial u}{\partial x}(t,x)=K(t), \qquad \frac{\partial u}{\partial t}(t,x)=\dot K(t)x+\dot v(t),$$9, $\Delta u$0, and intermediate-time decay constraints

$\Delta u$1

for all $\Delta u$2 in a nested set $\Delta u$3. For a fixed candidate hold $\Delta u$4, feasibility of the corresponding linear inequalities determines whether that inter-execution time is admissible; the MAC law selects the largest feasible $\Delta u$5 (Donkers et al., 2011).

The RL-triggered predictive architecture is structurally different. It is explicitly three-part: a high-level MPC plan generator, a low-level linear state-feedback compensator, and a recomputation policy learned by RL. At recomputation instants, the MPC produces a predicted trajectory and control sequence

$\Delta u$6

Between recomputations, the plant input is

$\Delta u$7

with saturation

$\Delta u$8

At each sampling instant, a binary action $\Delta u$9 either triggers a new MPC solve or continues execution of the current plan with the LQR compensator (Bøhn et al., 2020).

The 2025 receding-horizon MAMPC paper retains the standard discrete-time LTI model

$a_k:=1/h_k$0

and the quadratic tracking-plus-input-change stage cost

$a_k:=1/h_k$1

What distinguishes MAMPC is the additional $a_k:=1/h_k$2-type attention budget on successive differences, together with the inclusion of previous optimal inputs in the augmented horizon

$a_k:=1/h_k$3

This construction makes the input-change count horizon-consistent across receding-horizon shifts (Teja et al., 28 Jul 2025).

The Brockett-style formulation can also be read in predictive terms. The paper presents a finite-horizon optimal control functional in the Liouville setting, with nonlinear dynamics

$a_k:=1/h_k$4

state-density evolution

$a_k:=1/h_k$5

and a structured controller $a_k:=1/h_k$6. Although the paper is not MPC in the standard discrete-time sense, the finite-horizon, repeatedly solvable structure and explicit attention functional make it directly relevant to MAMPC interpretations (Lee et al., 2019).

4. Solution methods and computational mechanisms

For LP-based MAC/AAC, the computational mechanism is a feasibility search over admissible inter-execution times. For each $a_k:=1/h_k$7, the controller enforces all eCLF inequalities at the intermediate subintervals $a_k:=1/h_k$8, together with the input bound

$a_k:=1/h_k$9

The online problem is therefore a sequence of small LP feasibility tests, or, in the AAC case, a sequence of feasibility tests over a finite set of admissible decay rates. The paper explicitly states that it does not encode hold selection with binary variables; a MILP encoding is possible but not used (Donkers et al., 2011).

For sparse-input-change MAMPC, the nonconvex problem $h$0 is relaxed by introducing an auxiliary variable $h$1, where $h$2 is a block upper-bidiagonal difference operator mapping the stacked augmented input vector $h$3 to its successive differences. The relaxed problem $h$4 replaces the hard equality with a quadratic penalty

$h$5

and retains the sparsity constraint $h$6. The resulting alternating minimization has two steps: an outer convex quadratic program in $h$7 under linear dynamics and constraints, and an inner analytical sparsity projection that keeps only the $h$8 largest-magnitude components of $h$9 (Teja et al., 28 Jul 2025).

The RL-triggered update-interval formulation uses an augmented Markov state

$$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$0

where $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$1 is the last MPC recomputation time and $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$2 is the current time. In practice, the state vector is augmented with squared versions of each component and normalized before entering the policy. The action space is binary, and the policy is logistic regression: $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$3 Training uses GPOMDP with $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$4, episodic horizon $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$5, and learning rate $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$6. The stochastic policy itself provides exploration, and initialization favors $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$7, i.e. near-always recompute, before gradually learning to skip recomputations (Bøhn et al., 2020).

The Brockett-style minimum-attention framework derives first-order optimality conditions from an augmented Lagrangian involving the state density $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$8, an adjoint field $$\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2$$9, and the control $\mathcal X\times[0,T]$0. Stationarity with respect to $\mathcal X\times[0,T]$1 yields

$\mathcal X\times[0,T]$2

After imposing the structured controller $\mathcal X\times[0,T]$3, the paper adapts a one-shot method and proposes an iterative gradient algorithm that updates $\mathcal X\times[0,T]$4 and $\mathcal X\times[0,T]$5 with line search, using Liouville simulation, adjoint back-propagation by advection, and second-order finite differences in time (Lee et al., 2019).

5. Stability guarantees, feasibility, and theoretical status

The strongest formal guarantees in the cited literature appear in the eCLF-LP line. For single-hold CLF conditions and for the extended multi-subinterval eCLF conditions, the paper gives global exponential stability (GES) results of the form

$\mathcal X\times[0,T]$6

The MAC theorem states that, under the eCLF assumptions, selecting the largest feasible $\mathcal X\times[0,T]$7 and corresponding $\mathcal X\times[0,T]$8 yields GES with rate $\mathcal X\times[0,T]$9 and a gain bound $a\in\{0,1\}$0. The AAC theorem gives an analogous guarantee when the hold time is scheduler-provided and the controller chooses the largest feasible decay rate (Donkers et al., 2011).

The Brockett-style paper provides a different theoretical result: existence of a minimizer under structural assumptions. Specifically, with bounded $a\in\{0,1\}$1, $a\in\{0,1\}$2 continuously differentiable in $a\in\{0,1\}$3, and the controller restricted to $a\in\{0,1\}$4 with $a\in\{0,1\}$5, there exists a uniformly convergent minimizing sequence whose limit $a\in\{0,1\}$6 minimizes the functional $a\in\{0,1\}$7. A separate proposition states that the algorithm converges to a local minimizer for any initial control if the second variation of $a\in\{0,1\}$8 with respect to $a\in\{0,1\}$9 is degenerate (Lee et al., 2019).

The RL-triggered predictive architecture is explicitly more empirical in theoretical status. The paper does not provide formal proofs of recursive feasibility, ISS, or Lyapunov stability for the overall MPC/LQR/RL closed loop. Its robustness comes instead from the low-level linear compensator, the feasibility of the MPC subproblem, input constraints, and the practical possibility of imposing maximum recomputation intervals. The architecture is described as having a dual-mode flavor reminiscent of classical robust or dual-mode MPC, but formal terminal sets or robust tube tightening are not enforced (Bøhn et al., 2020).

The sparse-input-change MAMPC formulation is also explicit about its theoretical limits. The relaxed problem $\|\Delta u\|_0$0 is less conservative than the original $\|\Delta u\|_0$1, since any feasible solution of $\|\Delta u\|_0$2 is feasible for $\|\Delta u\|_0$3, but the alternating scheme is a heuristic: it “may not solve $\|\Delta u\|_0$4,” and $\|\Delta u\|_0$5 “may also result in a solution that is different from $\|\Delta u\|_0$6.” The paper does not provide KKT conditions, global convergence guarantees, or robust extensions for disturbances and model mismatch (Teja et al., 28 Jul 2025).

6. Representative applications and empirical behavior

The empirical studies show that minimum-attention formulations can substantially reduce updates, MPC solves, or input changes, but the performance trade-off depends strongly on the system and the chosen notion of attention.

Study	Setup	Reported outcome
Cart–pendulum	nonlinear plant, $\|\Delta u\|_0$7, RL recomputation penalty $\|\Delta u\|_0$8	learned policy outperforms baselines by $\|\Delta u\|_0$9; virtually same angle-stabilization as standard MPC with less than one fourth of the computational cost
Battery storage	$u(t,x)=v(t)+K(t)x,$00, $u(t,x)=v(t)+K(t)x,$01, $u(t,x)=v(t)+K(t)x,$02, $u(t,x)=v(t)+K(t)x,$03	RL policy outperforms baselines by $u(t,x)=v(t)+K(t)x,$04 in undiscounted return
Quadruple tank	$u(t,x)=v(t)+K(t)x,$05, $u(t,x)=v(t)+K(t)x,$06, $u(t,x)=v(t)+K(t)x,$07, $u(t,x)=v(t)+K(t)x,$08	sparse density falls from $u(t,x)=v(t)+K(t)x,$09 for MPC to $u(t,x)=v(t)+K(t)x,$10 for MAMPC $u(t,x)=v(t)+K(t)x,$11 over full run
SOFC stack	same horizons, $u(t,x)=v(t)+K(t)x,$12	sparse density reduced relative to MPC, but tracking error higher for MAMPC

In the cart–pendulum example, the MPC uses horizon $u(t,x)=v(t)+K(t)x,$13, sample period $u(t,x)=v(t)+K(t)x,$14, input constraints $u(t,x)=v(t)+K(t)x,$15, objective $u(t,x)=v(t)+K(t)x,$16 with $u(t,x)=v(t)+K(t)x,$17, and disturbance

$u(t,x)=v(t)+K(t)x,$18

Against the baselines “always recompute,” “never recompute,” and fixed periodic recomputation, the learned RL policy achieves virtually the same angle-stabilization performance as standard MPC while using less than one fourth of the computational cost, corresponding to approximately $u(t,x)=v(t)+K(t)x,$19 fewer MPC solves (Bøhn et al., 2020).

In the battery-storage example, the control problem is economic rather than purely regulatory. The state of charge satisfies

$u(t,x)=v(t)+K(t)x,$20

with $u(t,x)=v(t)+K(t)x,$21, $u(t,x)=v(t)+K(t)x,$22, $u(t,x)=v(t)+K(t)x,$23, input bounds

$u(t,x)=v(t)+K(t)x,$24

and horizon $u(t,x)=v(t)+K(t)x,$25. The paper reports that computational-energy cost is relatively small, so the main gains arise from learning when it is preferable to execute more of a single MPC plan under noisy forecasts rather than recomputing frequently (Bøhn et al., 2020).

The 2025 MAMPC paper reports detailed sparse-density and tracking-error comparisons. For the quadruple-tank system over the full run, MAMPC with $u(t,x)=v(t)+K(t)x,$26 gives $u(t,x)=v(t)+K(t)x,$27, $u(t,x)=v(t)+K(t)x,$28, $u(t,x)=v(t)+K(t)x,$29; MAMPC with $u(t,x)=v(t)+K(t)x,$30 gives $u(t,x)=v(t)+K(t)x,$31, $u(t,x)=v(t)+K(t)x,$32, $u(t,x)=v(t)+K(t)x,$33; and standard MPC gives $u(t,x)=v(t)+K(t)x,$34, $u(t,x)=v(t)+K(t)x,$35, $u(t,x)=v(t)+K(t)x,$36. After removing the first 15 steps, the reported MAMPC tracking errors become $u(t,x)=v(t)+K(t)x,$37 for $u(t,x)=v(t)+K(t)x,$38 and $u(t,x)=v(t)+K(t)x,$39 for $u(t,x)=v(t)+K(t)x,$40, versus $u(t,x)=v(t)+K(t)x,$41 for MPC (Teja et al., 28 Jul 2025).

For the SOFC stack, over the full run the paper reports MAMPC $u(t,x)=v(t)+K(t)x,$42: $u(t,x)=v(t)+K(t)x,$43, $u(t,x)=v(t)+K(t)x,$44, $u(t,x)=v(t)+K(t)x,$45; MAMPC $u(t,x)=v(t)+K(t)x,$46: $u(t,x)=v(t)+K(t)x,$47, $u(t,x)=v(t)+K(t)x,$48, $u(t,x)=v(t)+K(t)x,$49; MPC: $u(t,x)=v(t)+K(t)x,$50, $u(t,x)=v(t)+K(t)x,$51, $u(t,x)=v(t)+K(t)x,$52. With transients removed, the MAMPC sparse density remains below MPC, but tracking error remains higher, illustrating the expected trade-off between sparse control actions and tracking performance under tight attention budgets (Teja et al., 28 Jul 2025).

The earlier LP-based MAC paper also reports numerical advantages. In the batch-reactor example, with target $u(t,x)=v(t)+K(t)x,$53, $u(t,x)=v(t)+K(t)x,$54, and

$u(t,x)=v(t)+K(t)x,$55

minimum attention control achieves the required GES rate and significantly larger inter-execution times than a self-triggered implementation using $u(t,x)=v(t)+K(t)x,$56, while maintaining comparable Lyapunov decay (Donkers et al., 2011).

7. Design trade-offs, misconceptions, and open directions

A recurrent misconception is that minimum attention necessarily means event-triggered control in the narrow sense of threshold-based triggering. The cited RL-triggered MPC work is event-triggered in the broad sense that recomputation occurs conditionally at each sampling instant, but the trigger is learned from an augmented state rather than hand-crafted from error thresholds. The same paper explicitly distinguishes its approach from formulations in which update times are optimized inside the MPC problem; here, update timing is optimized by a separate RL policy interacting with the MPC and the plant (Bøhn et al., 2020).

A second misconception is that MAMPC always minimizes the number of input moves. That statement is exact only for the receding-horizon $u(t,x)=v(t)+K(t)x,$57-budgeted formulation, where “attention” is the zero norm of successive input differences. In the LP-based formulation, attention is $u(t,x)=v(t)+K(t)x,$58; in the Brockett-style formulation, attention is the integrated variability of the control map; and in the RL-triggered formulation, attention is the cost or energy of plan recomputation (Teja et al., 28 Jul 2025, Donkers et al., 2011, Lee et al., 2019, Bøhn et al., 2020).

The principal design trade-off is consistent across the formulations: reducing attention tightens the feasible set of control behaviors. In the sparse-input-change formulation, smaller $u(t,x)=v(t)+K(t)x,$59 yields fewer changes but can degrade tracking and feasibility; larger $u(t,x)=v(t)+K(t)x,$60 approaches standard MPC behavior. The weight $u(t,x)=v(t)+K(t)x,$61 penalizes the magnitude of allowed changes, while $u(t,x)=v(t)+K(t)x,$62 determines how tightly $u(t,x)=v(t)+K(t)x,$63 is enforced in the relaxed problem. In the RL-triggered architecture, larger MPC horizons provide longer plans that can support fewer recomputations, but they also increase the cost of each MPC solve. In the LP-based framework, a smaller $u(t,x)=v(t)+K(t)x,$64 improves the GES gain bound $u(t,x)=v(t)+K(t)x,$65 but increases the number of linear constraints and may cause infeasibility (Teja et al., 28 Jul 2025, Bøhn et al., 2020, Donkers et al., 2011).

The literature also delineates several open directions. The RL-triggered work identifies solver latency and self-triggering as future concerns, suggesting prediction of the next recomputation interval $u(t,x)=v(t)+K(t)x,$66 so that solver time can be scheduled explicitly. The Brockett-style nonlinear framework is computationally intensive and, as stated, is not yet efficient for real-time control without further approximation. The sparse-input-change MAMPC formulation does not yet address disturbances or model mismatch through robust design. This suggests that future MAMPC research will likely continue to combine predictive structure with explicit resource-aware objectives, while seeking stronger recursive feasibility, robustness, and real-time guarantees (Bøhn et al., 2020, Lee et al., 2019, Teja et al., 28 Jul 2025).