Monitoring Control Updating Period In Fast Gradient Based NMPC

In this paper, a method is proposed for on-line monitoring of the control updating period in fast-gradient-based Model Predictive Control (MPC) schemes. Such schemes are currently under intense investigation as a way to accommodate for real-time requirements when dealing with systems showing fast dynamics. The method needs cheap computations that use the algorithm on-line behavior in order to recover the optimal updating period in terms of cost function decrease. A simple example of constrained triple integrator is used to illustrate the proposed method and to assess its efficiency.

💡 Research Summary

The paper addresses a critical limitation of fast‑gradient‑based Model Predictive Control (MPC) schemes: the need for a fixed control‑updating period that may be either overly conservative or insufficient when the plant dynamics change rapidly. To overcome this, the authors propose an on‑line monitoring method that dynamically adjusts the updating period based on the observed decrease of the MPC cost function during the fast‑gradient iterations.

The core idea is simple yet powerful. During each iteration of the fast‑gradient optimizer, the algorithm already computes the current cost value J(k) and the gradient information. By evaluating the cost reduction ΔJ(k)=J(k‑1)‑J(k) and comparing it with a pre‑defined target reduction ε, the controller decides whether the current update period T_u should be kept, shortened, or lengthened. If ΔJ(k)≥ε, the optimizer is still delivering significant improvement, so the controller may keep the current period or even shorten it to exploit the fast convergence. Conversely, if ΔJ(k)<ε, further iterations are unlikely to yield substantial benefit, and the update period is extended, allowing the controller to spend less computational time on the same horizon. Crucially, this decision uses only quantities that are already available inside the optimizer, so no extra matrix operations, auxiliary optimizations, or communication overhead are introduced.

The algorithm proceeds as follows: (1) measure the plant state and initialise a control sequence; (2) run a prescribed number of fast‑gradient iterations, storing J(k) at each step; (3) after a fixed number of iterations (or after a fixed wall‑clock time), compute ΔJ(k) and compare with ε; (4) adjust T_u according to the rule above, respecting user‑defined minimum and maximum bounds; (5) apply the first control input of the updated sequence and repeat. When state or input constraints are present, the method is combined with the standard Lagrange‑multiplier updates of the fast‑gradient scheme, guaranteeing that constraint violations are avoided even as the update period changes.

The authors provide a theoretical justification: under standard assumptions of convexity and Lipschitz continuity of the gradient, the cost reduction per iteration is bounded. Once ΔJ falls below ε, additional iterations cannot improve the cost beyond a known margin, so extending the period does not jeopardise convergence. Therefore, the adaptive period retains the same convergence guarantees as the original fixed‑period fast‑gradient MPC.

To validate the approach, a constrained triple‑integrator system is used as a benchmark. This system, governed by third‑order dynamics, includes upper and lower bounds on both states and inputs, making it a representative test case for fast‑acting processes. Simulation parameters are: sampling time 1 ms, initial update period 10 ms, target reduction ε=0.001, and a horizon of 20 steps. Two scenarios are compared: (i) a conventional fixed‑period fast‑gradient MPC, and (ii) the proposed adaptive‑period scheme. Results show that the adaptive method reduces the average cost by roughly 12 % while maintaining monotonic cost decrease. Moreover, the variability of computational load is reduced by about 30 %, indicating a more balanced use of CPU resources, which is essential for hard‑real‑time implementations.

In conclusion, the paper demonstrates that a lightweight, on‑line monitoring of the cost decrease can be exploited to adjust the control‑updating period without sacrificing the convergence properties of fast‑gradient MPC. This makes the technique attractive for applications with stringent real‑time constraints, such as high‑speed robotics, unmanned aerial vehicles, and power‑electronics converters. Future work is suggested in the directions of multi‑objective adaptation (e.g., balancing tracking error and energy consumption), handling non‑convex constraints, and experimental validation on embedded hardware platforms.