Technical Summary: Suboptimal and Reduced-Order MPC via Timescale Separation

Problem Statement

Model Predictive Control (MPC) faces a persistent challenge in real-time implementation: the computational complexity of solving optimal control problems for high-dimensional or highly nonlinear systems within strict sampling intervals. While increased computational power has alleviated some burdens, solving these problems to global optimality in real-time remains infeasible for many applications.

To address this, two common strategies are often employed in practice:

1. Model Order Reduction: Utilizing simplified models that neglect certain dynamics (e.g., fast unmodeled dynamics or low-level compensator states) to reduce the dimensionality of the optimization problem.
2. Suboptimal Approaches: Computing only an approximate solution to the optimal control problem within a finite number of iterations, rather than waiting for convergence to the exact optimum.

However, rigorously analyzing the stability and feasibility of a closed-loop system that combines both strategies is difficult. The interplay between optimization errors (due to suboptimality) and model mismatch (due to reduced-order modeling) can destabilize the system. Existing literature often treats these strategies separately or relies on discrete-time formulations that do not explicitly account for the continuous-time nature of the physical plant and the role of the sampling time.

Methodology: The SMART-MPC Framework

The authors propose a generalized framework termed SuboptiMAl and Reduced-order Tunable MPC (SMART-MPC). This framework integrates suboptimal optimization and reduced-order modeling within a rigorous stability analysis based on timescale separation.

System Modeling

The physical plant is modeled as a continuous-time interconnected nonlinear system consisting of:

• Target Dynamics ($x_c$): The primary states of interest that enter the cost function.
• Extra Dynamics ($\xi_c$): Auxiliary states representing unmodeled dynamics or low-level compensators (e.g., electrical dynamics in a motor). These are assumed to be fast and exponentially stable, converging to an equilibrium manifold $\xi_{eq}(x_c, u_c)$ when the target state and input are fixed.
• Tunable Parameter ($\epsilon$): A parameter scaling the time constant of the extra dynamics, allowing for arbitrary acceleration of these fast dynamics.

The system is discretized with a sampling time $\delta$, which is treated not just as a fixed parameter but as a tunable design variable.

The Control Architecture

The SMART-MPC algorithm operates as follows:

1. Reduced-Order Model: The MPC optimizer uses a reduced-order model derived by constraining the extra dynamics to their equilibrium manifold ($\xi_c = \xi_{eq}(x_c, u_c)$). This eliminates the fast states from the optimization variables.
2. Suboptimal Optimization: At each sampling instant, the finite-horizon optimal control problem is solved suboptimally. The optimizer runs a generic algorithm (e.g., Sequential Quadratic Programming) for a finite number of iterations, updating an internal optimizer state $z_t$.
3. Closed-Loop Interconnection: The full-order physical plant (including the fast extra dynamics) is interconnected with the iterative optimizer. The control input is applied to the full plant, while the optimizer only "sees" the reduced model.

Theoretical Analysis

The core of the methodology relies on singular perturbation theory and timescale separation. The authors analyze the closed-loop system by decomposing it into:

• Fast Subsystem (Boundary Layer): The dynamics of the extra states and the optimizer state, assuming the target states are fixed.
• Slow Subsystem (Reduced System): The dynamics of the target states, assuming the fast states and optimizer are at their equilibrium manifolds.

By treating the sampling time $\delta$ as a tunable parameter, the authors prove that if $\delta$ is sufficiently small (and the extra dynamics are sufficiently fast relative to the sampling rate, i.e., $\epsilon = r\delta$), the closed-loop system behaves as a stable two-time-scale system.

Key Contributions

1. Unified Framework: The paper provides a rigorous methodological perspective that simultaneously handles suboptimal optimization and model order reduction, a combination often used heuristically in practice but rarely analyzed together.
2. Sampling Time as a Design Parameter: Unlike previous works that treat sampling time as fixed, this framework explicitly analyzes the role of $\delta$. It demonstrates that operating at a sufficiently fast sampling rate is the key to ensuring stability despite the use of reduced models and suboptimal solutions.
3. Formal Guarantees: The authors prove that under specific conditions (sufficiently small $\delta$ and appropriate tuning of $\epsilon$), the closed-loop system achieves:
   • Recursive Feasibility: The optimization problem remains feasible at every step.
   • Exponential Stability: The system converges exponentially to an equilibrium point where the target dynamics reach the desired setpoint, the optimizer state aligns with the optimal input sequence, and the extra dynamics reside on their equilibrium manifold.
4. Continuous-Time Foundation: The analysis is derived directly from continuous-time dynamics, bridging the gap between physical reality and discrete-time MPC implementation, offering a more rigorous foundation than purely discrete-time analyses.

Results and Validation

The theoretical results are validated through virtual experiments on an underactuated two-link robotic arm (PenduBot) driven by a DC motor, simulated in the high-fidelity MuJoCo physics engine.

• Experimental Setup: The MPC optimizer uses a reduced-order model that neglects the electrical dynamics (armature current) of the motor, relying instead on the mechanical dynamics. The optimization is performed using a single iteration of a Sequential Quadratic Programming (SQP) method per sampling instant (Real-Time Iteration).
• Performance with Fast Sampling: When the sampling time $\delta$ is small (0.005 s) and the electrical dynamics are fast relative to the sampling (small $\epsilon$), the SMART-MPC performance matches that of two ideal benchmarks: (1) suboptimal MPC with exact (full-order) prediction, and (2) optimal MPC with exact prediction. The system successfully performs a swing-up maneuver to the upright equilibrium.
• Performance Degradation: When the sampling rate is slowed (0.07 s) or the timescale separation is reduced (larger inductance $L$), the performance degrades, exhibiting oscillations and potential convergence failure. This confirms the theoretical necessity of sufficient timescale separation.
• Robustness: Monte Carlo analysis with random initial conditions demonstrates that the approach maintains stability and successfully stabilizes the robot in all trials under the fast-sampling regime.

Significance and Claims

The paper claims that the proposed SMART-MPC framework offers a theoretically grounded solution for deploying MPC on complex, high-dimensional systems where computational resources are limited. By leveraging timescale separation, the authors demonstrate that it is possible to:

• Neglect fast unmodeled dynamics without sacrificing stability.
• Use suboptimal solutions to reduce computational load without compromising recursive feasibility.
• Guarantee exponential stability and recursive feasibility through the proper tuning of the sampling time.

The authors emphasize that their work extends previous preliminary versions by providing rigorous proofs starting from continuous-time dynamics and explicitly characterizing the role of the sampling time. The validation on a challenging robotic setup in a high-fidelity simulator suggests the practical viability of the approach for real-world applications involving interconnected mechanical and electrical systems.