Multi-Rate NMPC: Multi-Time Scale Control

• MR-NMPC is a predictive control formulation that explicitly handles nonlinear dynamics and heterogeneous actuation by using different update rates in a unified model.
• It encompasses various methodologies including unified heterogeneous-actuation models, sampled-data lifted approaches, hierarchical planner-tracker schemes, and variational macro/micro discretizations.
• By aligning prediction horizons with distinct time scales, MR-NMPC improves robustness, constraint satisfaction, and computational efficiency in complex, multiscale systems.

Searching arXiv for the cited MR-NMPC-related papers to ground the article and verify bibliographic details. Multi-Rate Nonlinear Model Predictive Control (MR-NMPC) denotes a family of predictive-control formulations in which nonlinear dynamics, decision variables, or feedback updates are treated on more than one time scale rather than collapsed into a single-rate discretization. In the recent literature, this designation covers several distinct but related constructions: a unified MPC that embeds heterogeneous actuator bandwidths in one prediction model for a jet-powered humanoid (Gorbani et al., 22 May 2025); sampled-data NMPC with nonlinear lifting and fast-sample fast-hold approximations, which admits faster actuation than the measurement/update rate (Gerdpratoom et al., 10 Jan 2025); hierarchical planner-tracker architectures in which a slower MPC layer is paired with a faster CLF-, CBF-, or WBC-based execution layer (Csomay-Shanklin et al., 2022, Rosolia et al., 2020, Chunawala et al., 2 Jul 2026); and multirate variational NMPC that resolves fast internal dynamics on a micro grid while optimizing on a coarser macro grid (Lishkova et al., 2021). Across these formulations, the common objective is to align the predictive model and receding-horizon optimization with actuator bandwidth mismatch, intersample behavior, hybrid contact switching, or multiscale mechanics, rather than treating those effects as implementation afterthoughts.

1. Conceptual scope and taxonomy

The cited literature realizes MR-NMPC in several recurring forms. In a unified heterogeneous-actuation formulation, one optimizer includes fast and slow control channels directly in a single state-space prediction model and constrains them to update at different rates; the iRonCub controller exemplifies this by optimizing joint positions and jet throttle-related variables together while respecting the distinct update periods of joints and jet engines (Gorbani et al., 22 May 2025). In a sampled-data lifted formulation, the control decision is made at a slower sampling rate, but the held input is allowed to vary multiple times within each interval through an upsampling factor $M$, yielding a multi-rate lifted NMPC when $M>1$ (Gerdpratoom et al., 10 Jan 2025). In hierarchical multi-rate architectures, a slow planner computes a reference, contact plan, or reduced-order trajectory, while a faster nonlinear execution layer enforces full-order dynamics and constraints; this pattern appears in MPC+CLF designs using Bézier curves, MPC+CLF-CBF constructions for safety-critical systems, and SRB-based locomotion planning coupled to a whole-body controller (Csomay-Shanklin et al., 2022, Rosolia et al., 2020, Chunawala et al., 2 Jul 2026). A fourth pattern is macro/micro discretization, in which the receding-horizon problem is solved on a coarse macro grid but fast states and controls are still resolved internally on a micro grid through a multirate variational model (Lishkova et al., 2021).

This diversity suggests that MR-NMPC is not a single algorithmic template. Rather, it is a structural response to a recurring modeling fact: nonlinear plants often possess state, actuator, or contact processes that evolve on materially different time scales. Some formulations treat multi-rate behavior as a property of the control inputs; others treat it as a property of the prediction model or the planner–tracker decomposition.

Paradigm	Representative papers	Hallmark
Unified single-optimizer MR-MPC	(Gorbani et al., 22 May 2025)	Fast joints and slow jets in one predictive model
Sampled-data lifted MR-NMPC	(Gerdpratoom et al., 10 Jan 2025)	Faster piecewise-constant actuation inside each sample
Hierarchical planner-tracker MR control	(Csomay-Shanklin et al., 2022, Rosolia et al., 2020, Chunawala et al., 2 Jul 2026)	Slow MPC planning, fast nonlinear execution
Variational macro/micro MR-NMPC	(Lishkova et al., 2021)	Coarse horizon with micro-scale physics resolution

A recurring misconception is that any reduction in optimization frequency qualifies as MR-NMPC. The paper "Accelerated Nonlinear Model Predictive Control by Exploiting Saturation" explicitly does not present MR-NMPC in the usual sense; it proposes a heuristic reuse of saturated open-loop inputs and is described more accurately as saturation-triggered multi-step reuse or event-triggered NMPC acceleration (Dyrska et al., 2020).

2. Dynamical modeling strategies

A central differentiator among MR-NMPC methods is the choice of predictive model used to expose slow and fast processes. In the iRonCub formulation, the predictive state is built from a linearised centroidal momentum model augmented with a second-order nonlinear jet engine model. The centroidal momentum is defined as

$$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$

and, after including jet thrust contributions, its rate is written in body coordinates as

$$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$

Here $A_{\mathrm{lin}}(s)$ and $A_{\mathrm{ang}}(s)$ depend on joint configuration because jet orientations and lever arms vary with limb pose. The jet dynamics themselves are modeled as

$$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$

so thrust is a dynamic state rather than an instantaneous control input (Gorbani et al., 22 May 2025). This modeling choice is specifically motivated by the fact that the jets are slow and nonlinear.

Sampled-data lifted MR-NMPC adopts a different abstraction. Instead of discretizing the plant once per sampling period and optimizing only pointwise states, it lifts each intersample segment into a function-valued discrete-time object. For nonlinear systems, the lifted dynamics are written as

$$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$

with causality restored by embedding the plant in a closed-loop sampled-data system with a strictly proper discrete-time controller. The associated NMPC problem then minimizes an integral stage cost over each interval,

$$\sum_{k=0}^{N-1}\int_0^T l(x[k](\theta),u[k](\theta))\,d\theta + p(x[N](0)),$$

which makes intersample behavior explicit in both prediction and cost (Gerdpratoom et al., 10 Jan 2025).

In locomotion-oriented MR-NMPC, hybrid contact structure enters the predictive model directly. For wall-supported bipedal locomotion of a quadrupedal robot, the high-level controller uses a single rigid body model with state

$$x:=\operatorname{col}(p,\dot p,\theta,\omega)\in\mathbb{R}^{12},$$

and augments it with stance-limb end-effector positions so that foot placements become part of the prediction dynamics. The slow contact-point updates are represented through increments $M>1$0, while the fast inputs are environment reaction forces (Chunawala et al., 2 Jul 2026).

In multiscale mechanical systems, the multirate variational approach derives the discrete model from the forced Lagrange–d’Alembert principle. Slow and fast coordinates are separated as $M>1$1, a macro grid with step $M>1$2 is introduced, and each macro interval is subdivided into $M>1$3 micro steps so that

$M>1$4

Slow variables are represented on the macro grid, fast variables on the micro grid, and the multirate discrete Lagrangian is formed by summing micro-step contributions over one macro interval (Lishkova et al., 2021).

3. Multi-rate decision variables and horizon construction

The most direct expression of MR-NMPC is the use of decision variables with different update rules. In the iRonCub controller, the platform has joints controlled at $M>1$5 Hz in the low-level system, jet engines controlled at $M>1$6 Hz, and an MPC running at $M>1$7 Hz. The formulation introduces a rate ratio

$M>1$8

where $M>1$9 is the slower actuator frequency, and holds the slow input constant between update instants rather than recomputing it every MPC step as if the actuator could respond that quickly. When an MPC iteration is not aligned with the slow actuator’s sampling period, the optimizer constrains the first slow input to equal the previously applied one, $$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$0 (Gorbani et al., 22 May 2025). The significance is not merely scheduling; it is that the prediction model is forced to respect the actuator bandwidth actually available.

Sampled-data lifted NMPC expresses the same principle through faster input variation inside a slower outer loop. With upsampling factor $$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$1, the held input within one sampling interval is

$$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$2

Thus $$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$3 yields standard single-rate zero-order hold, whereas $$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$4 yields a multi-rate lifted NMPC in which actuation is updated multiple times inside a sampling interval (Gerdpratoom et al., 10 Jan 2025). This construction is specifically intended to improve intersample performance when the state measurement and optimization cycle are relatively slow.

In hybrid locomotion, slow decisions are coupled to contact switching rather than to a fixed lower actuator bandwidth. The slow input is allowed to change only at switching instants $$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$5, and its dynamics are written as

$$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$6

where $$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$7 is an indicator that is $$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$8 on switching instants and $$h=\begin{bmatrix} h^p \ h^w \end{bmatrix} =\begin{bmatrix} m\,{}^{J}v_{J,G} \ \mathbb{I}\,{}^{J}\omega_o \end{bmatrix}\in\mathbb{R}^6,$$9 otherwise (Chunawala et al., 2 Jul 2026). Here, multi-rate behavior reflects the physical semantics of foothold changes: contact points are piecewise constant between gait transitions, whereas forces evolve at every sample.

The variational MR-NMPC literature constructs multi-rate prediction horizons differently. The controller updates on a coarse macro grid, while fast dynamics are resolved internally on a micro grid. This allows larger time steps in the prediction horizon than single-rate schemes while retaining fast-scale resolution inside each macro interval (Lishkova et al., 2021). This differs from simple move-blocking because the fast subsystem is still represented in the model.

4. Objective functions, constraints, and guarantees

MR-NMPC objectives are usually structured so that the multi-rate model is not the only source of robustness; offset rejection, tube tightening, or low-level invariance are typically added. In the iRonCub controller, an offset-free MPC-style objective augments the state with integral-like errors $$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$0 and $$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$1, with dynamics

$$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$2

and the cost penalizes position, momentum, orientation, input changes, and accumulated offsets (Gorbani et al., 22 May 2025). This is used to reduce steady-state offsets under disturbances and modeling mismatch.

Lifted sampled-data NMPC encodes constraints over the whole sampling interval rather than only at sample instants. The optimization requires

$$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$3

and approximates the continuous-time flow with fast-sample fast-hold subdivision, Runge–Kutta state propagation, and Simpson quadrature (Gerdpratoom et al., 10 Jan 2025). This directly addresses a limitation emphasized in that paper: conventional NMPC optimizes only at sampling instants and does not directly account for intersample trajectory behavior or constraint violations between samples.

Hierarchical MR-NMPC frameworks often secure guarantees by splitting responsibilities across time scales. In the Bézier-curve MPC+CLF architecture, the planner solves a finite-time optimal control problem on a linearized discrete model, while the low-level CLF-QP tracks a continuous reference trajectory on the full nonlinear system. State safety is enforced by requiring Bézier control points to lie in a tightened set $$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$4, where $$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$5 is a disturbance tube induced by the low-level tracking law, and the paper proves recursive feasibility, state safety, and input feasibility under the stated assumptions (Csomay-Shanklin et al., 2022). In the CBF-MPC hierarchy, sufficient conditions are given for low-level safety, low-level tracking, high-level safety, and high-level tracking, leading to recursive constraint satisfaction of state and input constraints for all time (Rosolia et al., 2020).

The quadrupedal wall-supported locomotion framework combines stage and terminal penalties on both reduced-order body states and slow contact-point states: $$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$6

$$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$7

$$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$8

subject to dynamics, input-availability constraints $$\dot{h}= \begin{bmatrix} -S({}^{B}\omega_B)\,h^p + A_{\mathrm{lin}}(s)T + mg\,{}^{B}R_J e_3 \ -S({}^{B}\omega_B)\,h^w + A_{\mathrm{ang}}(s)T \end{bmatrix}.$$9, $A_{\mathrm{lin}}(s)$0, and feasibility constraints $A_{\mathrm{lin}}(s)$1, $A_{\mathrm{lin}}(s)$2, $A_{\mathrm{lin}}(s)$3 (Chunawala et al., 2 Jul 2026).

A notable theme is that formal guarantees are uneven across the literature. Recursive feasibility and safety theorems are explicit in the hierarchical CBF/CLF-based works and in the multirate variational tube-MPC construction (Rosolia et al., 2020, Csomay-Shanklin et al., 2022, Lishkova et al., 2021), whereas the saturation-triggered acceleration method provides no new rigorous stability proof for the accelerated scheme itself and is justified empirically (Dyrska et al., 2020).

5. Computational structure and solver design

MR-NMPC is frequently motivated by online tractability. The iRonCub study linearizes the nonlinear augmented model online into a linear time-varying or LPV form, discretizes it with forward Euler, and solves the resulting MPC as a QP with OSQP. The controller runs at $A_{\mathrm{lin}}(s)$4 Hz, uses a variable sampling horizon so that a $A_{\mathrm{lin}}(s)$5-second horizon is represented with only $A_{\mathrm{lin}}(s)$6 knots, and reports an average solve time of $A_{\mathrm{lin}}(s)$7 ms, standard deviation $A_{\mathrm{lin}}(s)$8 ms, and maximum $A_{\mathrm{lin}}(s)$9 ms, which fits within the $A_{\mathrm{ang}}(s)$0 ms budget for $A_{\mathrm{ang}}(s)$1 Hz control (Gorbani et al., 22 May 2025).

Sampled-data lifted NMPC faces a different computational burden because the exact nonlinear flow map is generally unavailable in closed form. The proposed remedy is FSFH approximation plus numerical integration; the case studies use MATLAB fmincon for the Van der Pol oscillator and OpEn with PANOC for the inverted pendulum on a cart (Gerdpratoom et al., 10 Jan 2025). In hierarchical architectures, the computational burden is often shifted away from a monolithic nonlinear program. The CBF-MPC framework uses a convex low-level CLF-CBF QP and a convex high-level MPC when the nonlinear system is control affine and the planning model is linear (Rosolia et al., 2020). The wall-supported locomotion controller uses CasADi + IPOPT at $A_{\mathrm{ang}}(s)$2 Hz for the high-level MR-NMPC and a $A_{\mathrm{ang}}(s)$3 kHz low-level WBC; the reported mean solve time is $A_{\mathrm{ang}}(s)$4 ms with standard deviation $A_{\mathrm{ang}}(s)$5 ms (Chunawala et al., 2 Jul 2026).

The variational multirate approach addresses complexity by solving the receding-horizon problem only on macro nodes while retaining micro-level physical resolution in the model. Its online problem is a tube-based SOCP derived from successive linearization, and the numerical examples show substantial computational savings as the multirate factor $A_{\mathrm{ang}}(s)$6 increases, albeit with a performance–feasibility trade-off when $A_{\mathrm{ang}}(s)$7 becomes too large (Lishkova et al., 2021).

Two adjacent computational ideas define the boundary of MR-NMPC. First, LPV embedding plus SQP provides a QP-based iteration scheme for efficient NMPC, with approximate Hessians and constraint Jacobians to reduce online burden; the paper is not explicitly multi-rate, but it is presented as conceptually compatible with multi-rate control because expensive model or optimization updates could occur at one rate and control-law evaluation at another (Karachalios et al., 2024). Second, saturation-triggered reuse skips NLP solves when the predicted open-loop input remains at a bound, effectively lowering the update rate during saturated intervals, but only in a data-dependent, heuristic fashion rather than through a fixed multi-rate model or architecture (Dyrska et al., 2020).

6. Applications, empirical evidence, and recurrent limitations

The applications represented in the cited literature span aerial humanoids, sampled-data nonlinear benchmarks, safety-critical balancing, multiscale mechanics, and hybrid legged locomotion. In the iRonCub simulations in MuJoCo at $A_{\mathrm{ang}}(s)$8 Hz, the robot is subjected at $A_{\mathrm{ang}}(s)$9 s to a disturbance consisting of a $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$0 torque about the $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$1-axis and a $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$2 force along the $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$3-axis for $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$4 s, and successfully recovers despite approximately $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$5 roll tilt, about $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$6 m displacement in $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$7, and about $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$8 m vertical drop. In minimum-jerk trajectory tracking, roll and yaw MAE are below $$\ddot T_i = h(T_i,\dot T_i) + g(T_i,\dot T_i)\,v(u_{\mathrm{th},i}),$$9 rad and pitch MAE is about $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$0 rad. Ablations show that omitting jet dynamics causes failure to stabilize and that removing multi-rate handling worsens orientation oscillations markedly; reported orientation MAE is substantially better for the multi-rate controller than the single-rate variant (Gorbani et al., 22 May 2025).

In lifted sampled-data NMPC, the inverted-pendulum study compares conventional NMPC, single-rate lifted NMPC, and multi-rate lifted NMPC for sampling periods $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$1 s with control period fixed at $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$2 s, corresponding to $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$3. The multi-rate lifted controller improves performance at $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$4 s, clearly outperforms the others at $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$5 s, and at $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$6 s successfully swing-ups and stabilizes the system while conventional NMPC and single-rate lifted NMPC fail (Gerdpratoom et al., 10 Jan 2025).

The wall-supported locomotion framework is validated in RaiSim on a Unitree A1 quadruped in a $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$7 cm narrow corridor with unknown rough terrain. Over $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$8 random rough-terrain trials, the proposed MR-NMPC achieves a $$x[k+1](\theta)=\phi((k+1)T+\theta,\,x[k](T),\,u[k+1](\cdot)), \qquad y[k](\theta)=h(x[k](\theta)),$$9 higher success rate than conventional MPC with heuristic foot placement. Under a $$\sum_{k=0}^{N-1}\int_0^T l(x[k](\theta),u[k](\theta))\,d\theta + p(x[N](0)),$$0 N sinusoidal push from $$\sum_{k=0}^{N-1}\int_0^T l(x[k](\theta),u[k](\theta))\,d\theta + p(x[N](0)),$$1 s to $$\sum_{k=0}^{N-1}\int_0^T l(x[k](\theta),u[k](\theta))\,d\theta + p(x[N](0)),$$2 s, the heuristic baseline destabilizes, whereas the MR-NMPC remains feasible by selecting smaller, asymmetric steps (Chunawala et al., 2 Jul 2026).

In safety-critical and robustness-oriented hierarchical designs, the empirical emphasis is different. The Segway example shows that a low-frequency high-level MPC paired with a fast CLF-CBF controller can avoid overshoot or oscillation seen in baseline linear or nonlinear MPC designs while using smaller convex problems (Rosolia et al., 2020). The Bézier-based MPC+CLF framework shows, on a constrained nonlinear benchmark with additive disturbance, that low-level CLF alone can violate state or input constraints, MPC alone can violate constraints because it does not robustly account for the nonlinear tracking dynamics, and the proposed multi-rate construction satisfies both state and input constraints for all time (Csomay-Shanklin et al., 2022). The variational multirate examples further show improved energy behavior relative to forward Euler and substantial computation-time reductions in the Fermi–Pasta–Ulam system (Lishkova et al., 2021).

Several limitations recur. Some methods remain simulation-only (Dyrska et al., 2020, Gerdpratoom et al., 10 Jan 2025, Chunawala et al., 2 Jul 2026). Some rely on nominal models without added robustification in disturbed tests (Dyrska et al., 2020). Some derive strong guarantees only under structural assumptions such as control-affine nonlinear dynamics, linear planning models, or feasible low-level QPs (Rosolia et al., 2020, Csomay-Shanklin et al., 2022). And in the multirate variational setting, overly large macro steps can degrade performance or feasibility (Lishkova et al., 2021). Taken together, these results indicate that MR-NMPC is most effective when the selected time-scale decomposition matches a genuine physical or algorithmic separation—slow propulsion versus fast pose actuation, footholds versus contact forces, planning versus execution, or macro-grid optimization versus micro-grid mechanics—rather than being imposed solely for convenience.