Nonlinear MPC for Feedback-Interconnected Systems: a Suboptimal and Reduced-Order Model Approach

1. Problem Statement

The paper addresses the computational complexity inherent in Nonlinear Model Predictive Control (MPC) for complex dynamical systems. Standard MPC requires solving an optimal control problem online at every sampling instant, which is often computationally prohibitive for high-dimensional or fast-sampling systems.

The authors focus on a specific class of discrete-time feedback-interconnected systems characterized by two distinct timescales:

1. Target Dynamics ($x_t$): The primary state to be controlled (slow dynamics).
2. Extra Dynamics ($\xi_t$): Faster dynamics, which may represent unmodeled effects, high-frequency subsystems, or low-level compensators.

The core challenge is to design a control architecture that:

• Uses a reduced-order model (ignoring the fast dynamics $\xi_t$) to simplify the optimization.
• Operates suboptimally (performing only a finite number of optimization iterations rather than solving to exact optimality) to reduce computational load.
• Guarantees stability of the full-order closed-loop system despite the model reduction and suboptimal control actions.

2. Methodology

A. System Modeling

The system is modeled as a discrete-time feedback interconnection:
$$\begin{aligned} x_{t+1} &= f(x_t, \xi_t, u_t, \delta) \\ \xi_{t+1} &= g(\xi_t, x_t, u_t, \delta) \end{aligned}$$
where $\delta > 0$ is a parameter representing the sampling time. The authors leverage timescale separation, where $\delta$ acts as the small parameter that accelerates the fast dynamics relative to the slow ones.

• Assumption 1: The fast dynamics are Lipschitz continuous with respect to the slow states and inputs, scaled by $\delta$.
• Assumption 2: The fast subsystem possesses a globally exponentially stable equilibrium manifold $\xi_{eq}(x, u)$, uniformly in $(x, u)$.

B. Control Strategy

The proposed architecture combines two efficiency strategies:

1. Reduced-Order Model: The MPC optimizer uses a simplified model $f_R$ that assumes the fast state $\xi$ is instantaneously at its equilibrium $\xi_{eq}(x, u)$.
   $$f_R(x, u, \delta) := f(x, \xi_{eq}(x, u), u, \delta)$$
2. Suboptimal Optimization: Instead of solving the finite-horizon optimal control problem to global optimality, the solver performs a fixed, finite number of iterations (e.g., one step of a projected gradient method). The optimizer state $z_t$ evolves according to a dynamics $A(z_t, x_t)$, converging to the optimal solution $z^*(x_t)$.

C. Stability Analysis Framework

The authors employ singular perturbation theory (timescale separation) adapted for discrete-time systems to prove stability. The analysis decomposes the closed-loop system into:

1. Boundary-Layer System (Fast Subsystem): Analyzes the convergence of the fast state $\xi_t$ and the optimizer state $z_t$ to their equilibrium manifolds, assuming the slow state $x_t$ is fixed.
2. Reduced System (Slow Subsystem): Analyzes the stability of the slow state $x_t$ assuming the fast states are perfectly at equilibrium and the optimizer is optimal.
3. Interconnected System: Combines the above results to show that if the sampling time $\delta$ is sufficiently small, the "drift" caused by the mismatch between the actual fast dynamics and the ideal equilibrium is small enough to preserve global exponential stability.

3. Key Contributions

• Unified Architecture: The paper proposes the first framework that simultaneously integrates suboptimal control and model order reduction for nonlinear feedback-interconnected systems.
• Rigorous Stability Proof: It establishes global exponential stability for the closed-loop system formed by the interconnection of the full-order plant and the suboptimal/reduced-order MPC optimizer.
• Sampling Time as a Stability Parameter: The authors formally characterize the sampling time $\delta$ not just as a discretization parameter, but as the critical timescale separation parameter. They prove that there exists a bound $\bar{\delta}$ such that for any $\delta < \bar{\delta}$, the system remains stable.
• Handling Model Mismatch: The theory explicitly accounts for the structural mismatch between the reduced-order model used in the optimizer and the full-order reality of the plant.

4. Results

The theoretical findings were validated through numerical simulations on a mechatronic system: a pendulum actuated by a DC motor.

• System Setup: The pendulum angle and velocity ($x$) are the slow states, while the armature current ($\xi$) represents the fast dynamics.
• Simulation Scenarios: The authors compared their proposed "Suboptimal and Reduced-Order" approach against:
  1. Optimal MPC with an Ideal Plant (no mismatch).
  2. Suboptimal MPC with an Ideal Plant.
  3. The proposed method (Suboptimal + Reduced-Order) interacting with the Full-Order Plant.
• Key Findings:
  • Small $\delta$ (0.01s): The proposed method achieved exponential convergence comparable to the ideal benchmarks, validating the timescale separation theory.
  • Medium $\delta$ (0.12s): Performance degraded for the proposed method when interacting with the full-order plant, while ideal benchmarks remained stable. This highlighted the necessity of the timescale separation condition.
  • Large $\delta$ (0.2s): The proposed method failed to stabilize the system, whereas the optimal MPC on the ideal plant still worked. This demonstrated that without sufficient timescale separation, the combined errors of model reduction and suboptimal optimization lead to instability.

5. Significance

This work is significant for the practical implementation of MPC in resource-constrained or high-speed applications (e.g., robotics, aerospace, power electronics) where:

1. Computational constraints prevent solving full-order, exact optimization problems in real-time.
2. System complexity involves fast unmodeled dynamics or low-level controllers that are difficult to model precisely.

By proving that stability can be guaranteed even when using a simplified model and an approximate solver—provided the sampling rate is fast enough relative to the system dynamics—the paper provides a theoretical foundation for deploying lightweight, efficient MPC controllers in real-world engineering systems. It bridges the gap between theoretical control guarantees and practical computational limitations.