Enhanced Sampled-Data Model Predictive Control via Nonlinear Lifting

This paper introduces a novel nonlinear model predictive control (NMPC) framework that incorporates a lifting technique to enhance control performance for nonlinear systems, addressing a gap where lifting has been widely employed in linear systems but its application to nonlinear systems remains unexplored. The authors formulate an NMPC scheme that combines fast-sample fasthold (FSFH) approximations and numerical methods to approximate system dynamics and cost functions.

The paper addresses the challenge of defining the nonlinear counterpart of the direct feedthrough term, which is absent in the original continuous-time system, noting that In the linear case, this effect can be described without involving state transitions, but for nonlinear systems, this is clearly not possible. The authors propose a solution by first utilizing fast-sample fast-hold approximation [3] alongside numerical integration methods to approximate solutions of the nonlinear ordinary differential equation.

The proposed approach overcomes limitations of standard NMPC formulations, which typically solve an optimization problem based on costs evaluated only at each sampling instant, and fails to incorporate intersample behavior or ensure constraints are satisfied between sampling instants. The authors present an ideal NMPC formulation that incorporates nonlinear lifting:

"Now, we present an ideal NMPC formulation that incorporates nonlinear lifting. Instead of (11), we address the following optimal control problem at each sampling instant t = nT, n ∈ Z:

minimise

v[k]∈Rpc, k=0,N–1

X

N–1

k=0

Z T 0

l(xk, uk)dθ + φ(xN)

subject to x0 = Φ(θ, x(nT), v[0])

xk + 1 = Φ((k + 1)T + θ, xk, H(v[k]))

xk ∈ X

uk ∈ U

 k = 0,...,N – 1

xN ∈ Xf"

To handle the explicit solution of the system dynamics, which is generally unobtainable, the authors employ a fast-sample fast-hold (FSFH) approximation by subdividing the sampling interval into segments and applying numerical methods such as Runge-Kutta methods to approximate solutions. For computing the integration in the cost function, a numerical integration method like Simpson’s rule is applied.

The effectiveness of this method is validated through two case studies:

1. The Van der Pol oscillator: Simulation results demonstrate that the lifted NMPC outperforms conventional NMPC in terms of reduced settling time and improved control accuracy.

2. The inverted pendulum on a cart: This study shows that the multi-rate lifted NMPC can effectively control the system even with a slow sampling period, where single-rate control fails, demonstrating superior performance compared to conventional NMPC and single-rate lifted NMPC, particularly at longer sampling periods.

The paper concludes that these results underscore the potential of the lifting-based NMPC framework and its multi-rate extension for practical applications requiring optimal control of nonlinear and highly dynamic systems. The method is computationally efficient due to the use of approximations like FSFH and numerical integration methods. The authors also present a multi-rate control scheme where the control frequency exceeds the sampling frequency, which more clearly demonstrates the advantage of the lifted NMPC.

Key findings include: Simulation results demonstrate that our method enhances performance in nonlinear control by effectively suppressing unwanted overshoot and undershoot. Furthermore, the multi-rate lifted NMPC framework ensures that control specifications are met even at slow sampling rates. The comparison across different sampling periods (T = 0.1, 0.25, and 0.5 s) shows the multi-rate approach achieving superior performance over conventional NMPC and single-rate lifted NMPC in terms of state trajectories and control inputs. For instance, at T = 0.5 (s), the multi-rate NMPC successfully achieves swing-up and stabilises the system, whereas the other two methods fail." The research offers a practical solution for real-time applications in nonlinear control.

The paper is supported by JSPS KAKENHI Grant Number JP24K07546. The key keywords are Nonlinear lifting, nonlinear model predictive control, sampled-data systems. The paper is available as arXiv:2501.05815v1 [eess.SY] on 10 Jan 2025.

The references include works related to lifting techniques in linear periodic systems and nonlinear sampled-data systems, such as [20], which the authors build upon to propose their specific nonlinear framework.

As a fastidious researcher, I have analyzed the provided paper on Enhanced sampled-data model predictive control via nonlinear lifting. The core contribution is a novel Nonlinear Model Predictive Control (NMPC) framework that incorporates a lifting technique to explicitly account for intersample dynamics in discrete-time systems, which are challenging to handle with standard NMPC formulations.

Here are the specific improvements and capabilities this lifted NMPC framework can enable for AI systems:

Improved AI System Capabilities via Lifted NMPC Framework:

1. Enhanced Control of Highly Nonlinear Physical Systems:

2. ⏱ Superior Settling Time and Transient Response in Dynamic Environments:

3. Robust Performance Under Slow Sampling Rates (Multi-Rate Capability):

4. Constraint Satisfaction Between Samples (Intersample Constraint Handling):

Specific Technical Improvements and Functionality:

1. Explicit Intersample Dynamics Modeling:

2. Optimized Optimal Control Sequence Generation:

3. Computationally Efficient Real-Time Implementation:

4. Enhanced Performance in Nonlinear Tracking Tasks (e.g., Robotics, Aerospace):

Detailed Breakdown of Improvements:

1. Explicit Intersample Dynamics Modeling:

2. The framework explicitly transforms a continuous-time nonlinear system into a discrete-time model that preserves intersample behavior via the lifting operator (Section 2.2).

3. This allows the NMPC cost function (Equation 12) to integrate the dynamics over the interval between sampling instants, ensuring that control actions are optimized not just at instant snapshots, but considering how they affect system evolution until the next sample time.

4. Optimized Optimal Control Sequence Generation:

5. The NMPC optimization problem is reformulated to minimize a cost function that includes an integral term over the lifting domain (Equation 12):

6. This ensures that the calculated control sequence optimizes performance across the entire sampling period, leading to smoother, more precise trajectories compared to conventional NMPC which only evaluates costs at sampling instants (Equation 11).

7. Computationally Efficient Real-Time Implementation:

8. The paper proposes using a Fast-Sample Fast-Hold (FSFH) approximation combined with numerical integration methods like Runge-Kutta for state approximation and Simpson's rule for cost integration.

9. This approach allows the complex, continuous system dynamics to be approximated numerically without requiring the explicit, often intractable, closed-form solution of the nonlinear differential equation (Section 5).

10. Enhanced Performance in Nonlinear Tracking Tasks (e.g., Robotics, Aerospace):

11. The validated case studies—the Van der Pol oscillator and the inverted pendulum on a cart—demonstrate that this method significantly reduces settling time and improves control accuracy compared to conventional NMPC, even under challenging constraints (Section 4).

12. Robust Performance Under Slow Sampling Rates (Multi-Rate Capability):

13. The framework can be extended to a multi-rate scheme where the controller operates at a higher frequency than the system's sampling rate (Equation 13).

14. This is particularly powerful for systems where slow sampling periods are necessary due to computational constraints, as demonstrated in the inverted pendulum example, where multi-rate control successfully stabilizes the system when single-rate methods fail (Section 4.2.2).

Summary of Impact:

The resulting AI system will be capable of performing high-precision, constrained trajectory tracking for complex nonlinear physical processes (like robotics or chemical processes) in real-time environments where the underlying dynamics are continuous but control decisions must be made discretely. It offers a significant advantage over existing NMPC methods by explicitly modeling and optimizing the system's behavior during the time interval between discrete measurements, leading to faster convergence, better constraint adherence, and stability at low sampling frequencies.