Model-Guided Local Bayesian Optimization for Tuning of Interpretable Controllers in Injection Molding

We propose a method to automatically optimize interpretable controllers during manufacturing while being cycle-efficient and risk-aware. The approach uses a Physics-Inspired Neural Mixture-of-Local-Experts model of the injection molding dynamics and augments its simulated closed-loop costs with a residual Gaussian Process, enabling Local Bayesian Optimization of controller parameters.

The method is central to a composite objective function that approximates closed-loop costs by combining two models:

1. a Physics-Inspired Neural Mixture-of-Local-Experts model that approximates the underlying process dynamics, and

2. a Gaussian Process (GP) regression model that corrects the mismatch between the dynamic model’s simulated closed-loop costs and real-world plant observations.

The algorithm computes new controller parameters during the cooling phase of the IMP, utilizing an acquisition function to balance predicted performance against uncertainty.

We benchmark the algorithm against Vanilla Bayesian Optimization (BO) in simulation, using three controllers with parameter counts ranging from 1 to 30. Using the local method, we identify controller parameters that yield costs comparable to or lower than those of global BO over 20 optimization iterations, while mitigating high-cost excursions during tuning.

The contribution is structured as follows. First, the dynamic model is introduced in Sec. II, which is then integrated into the trust-region optimization using Local BO in Sec. III. In Sec. IV, we define the tested controllers and the simulation setup to assess the effectiveness of MGLBO. Results are presented in Sec. V and discussed in Sec. VI.

The MGLBO algorithm is provided as pseudocode in Alg. 1, which utilizes a trust-region based approach, specifically employing the TurBO-1 algorithm [14] to find local optimal solutions within a defined trust region around the current best parameter vector θ∗C. The next evaluation point is determined by minimizing an acquisition function:

p(θ C) = E(Jtotal (θ C)) + κ Var(Jtotal (θ C)).

The performance of the controllers is assessed using three metrics:

1. We evaluate the controller performance using the minimum observed cost Jmin. This comparison is particularly useful for determining whether the local exploitation of MGLBO can achieve solution quality comparable to that of global optimization in Vanilla BO.

2. The maximum cost Jmax is used to evaluate the safety of the tuning process. Monitoring this metric is critical to industrial deployment, as unsafe controller parameters risk causing physical damage to the machine.

3. The metric of cumulative worsening JW (n) = n X max(0, Ji − Ji−1) i=2 is also important in industrial applications. If machine operators observe frequent deterioration in controller performance, they may disable the algorithm to protect the IM machine.

In Fig. 4, we visualize the distributions for the metrics from Sec. IV for both MGLBO and Vanilla BO, derived from 50 individual runs. It can be seen that for the MGLBO, all second controller generations yield only minor improvements over the first ones. During the MGLBO, all controllers converge to a final parametrization. The minimum costs obtained by MGLBO are lower than or equal to those of Vanilla BO, while the maximum costs correspond to the initial costs. The first deterioration of the controllers begins at iteration five, coinciding with the algorithm approaching convergence. For optimization using Vanilla BO, we observe that for the P controller, Vanilla BO shows an initial improvement higher than that of MGLBO. For all other controllers, the optimization converges more slowly compared to MGLBO. In Fig. 6, we visualize the control law obtained by optimizing the RBF controller as a function of nondimensional cavity pressure starting as a P controller. The control law again starts with a P controller, successively adding nonlinearity to π. For lower cavity pressures, the RBF controller increases the u+S;lim compared to the initial control law. For cavity-pressure values close to the cavity-pressure reference, the control law of the RBF controller approaches the initial proportional control law. The control law of the RBF controller exhibits a negative offset for pC = pref C, unlike the P control law, where the P controller output is exactly zero.

The MGLBO approach demonstrates strong potential for risk-aware, data-efficient controller optimization in the IMP. Adding model information enables a more targeted optimization than Vanilla BO, with lower maximum costs and lower cumulative worsening. For a higher number of parameters, such as the RBF controller, the advantages of the composite objective function also become apparent for Jmin, since the prediction of the costs using the dynamic model is not influenced by a higher parameter count NθC. The algorithm takes very small steps at the beginning of the optimization because the value-at-risk acquisition function is employed, and the GP model relies on the initial observation. This behavior could be improved by conducting additional prior experiments to obtain an initial set of observations, or by making a reasonable guess about the GP hyperparameters.

The proposed approach must be adapted for the industrial application. The transition points between the local expert models are not updated in this work and are assumed to be known. For a real-world application, these points must be updated, by e.g., using the Maximum-Expectation Algorithm [23] or an Interacting Multiple Model filter [24]. Although the drive model can capture nonlinear dynamics, in its current form, it is limited to single-state drives. While this may be sufficient for IMs with electric drives, modeling IMs with hydraulic drives may require adding additional states [10].

Future work will involve testing the algorithm on real IM hardware under realistic conditions and comparing different interpretable control laws with model-based controllers. Beyond guiding controller optimization, incorporating a process model into the optimization loop enables additional options. One could add process-dependent constraints, such as maximum admissible cavity-pressure overshoots. Combined with quality-attribute models that use the plant’s simulated pressure trajectories, we could also treat quality attribute references as equality constraints in the controller optimization. This could yield a tightly controlled IMP while meeting desired quality-attribute references.

The paper states: To our knowledge, this is the first work to propose an IM process-controller-optimization approach that leverages a dynamic process model and plant observations to achieve both data efficiency and risk awareness.

The paper also notes: "The contribution is structured as follows. First, the dynamic model is introduced in Sec. II, which is then integrated into the trust-region optimization using Local BO in Sec. III. In Sec. IV, we define the tested controllers and the simulation setup to assess the effectiveness of MGLBO."

The paper also mentions: "The approach uses a Physics-Inspired Neural Mixture-of-Local-Experts model of the injection molding dynamics and augments its simulated closed-loop costs with a residual Gaussian Process, enabling Local Bayesian Optimization of controller parameters."

And: We propose a method to automatically optimize interpretable controllers during manufacturing while being cycle-efficient and risk-aware.

The paper also states: We investigate this approach for three interpretable controllers: 1) Proportional (P) controller, 2) Gain-Scheduled Proportional Integral (G-PI) controller, and 3) Radial-Basis-Function (RBF) controller.

And: In simulations, we benchmark the performance of the MGLBO against Vanilla BO.

The paper also states: The control law for a simple feedback controller can be written as uS (k) = π(x(k), c(k), pref C (k), θ C) with the vector of controller parameters θ C ∈ RNθC, which we aim to optimize.

And: To measure the performance of the controller over one cycle of length NC, we define the cost function J= (13) k=2

J = 1/NC X [pC(k) − pref C(k)] squared + R uS(k) − uS(k-1)"

And: Combining the plant model and the controller allows predicting the costs JM, namely by inserting the simulated cavity-pressure curves and simulated controller outputs into (13).

And: "Since we do not assume that this simulated cost model perfectly matches the costs calculated from real plant experiments, we add a GP regression model to correct for this mismatch using previous observations. The sum of the deterministic model JM and the GP ˜ C) Jtotal (θ C) = JM (θ C) + J(θ ˜ C) ∼ GP 0, kSE (θ C, θ ′C)"

And: We denote the Squared Exponential (SE) kernel function with kSE.

The paper also states: The algorithm computes new controller parameters during the cooling phase of the IMP, utilizing an acquisition function to balance predicted performance against uncertainty.

And: We define the trust region at optimization iteration n to be S = θ C ∈ RNθC θ∗C − b ≤ θ C ≤ θ∗C + b, with the vector b defining the trust box around θ∗C, where the inequality holds element-wise.

And: "If the algorithm determines controller parameters that yield lower costs, we increase the trust-region size and move its center to the new best controller parameters. If it fails, we expect the model to be an inaccurate approximation to the real cost function and shrink S."

The paper also states: "To compare MGLBO against a baseline controller-tuning method, we use the MATLAB-internal bayesopt function with the expected-improvement-plus acquisition function [20]. For the Vanilla BO, we define bounds θ ± C for each controller to limit the search space."

And: We normalize all costs by the costs achieved by using the P controller with the initial gain KPInit.

And: "To assess MGLBO’s performance, we compare it using three metrics. 1) We evaluate the controller performance using the minimum observed cost Jmin. 2) The maximum cost Jmax is used to evaluate the safety of the tuning process. 3) The metric of cumulative worsening JW (n) = n X max(0, Ji − Ji−1) i=2 is also important in industrial applications."

And: The MGLBO approach demonstrates strong potential for risk-aware, data-efficient controller optimization in the IMP.

And: Adding model information enables a more targeted optimization than Vanilla BO, with lower maximum costs and lower cumulative worsening.

And: "For a higher number of parameters, such as the RBF controller, the advantages of the composite objective function also become apparent for Jmin, since the prediction of the costs using the dynamic model is not influenced by a higher parameter count NθC."

And: The Vanilla BO does not find lower minimum costs for P and G-PI. This may indicate that global optimization methods are not required to identify optimal, interpretable cavitypressure controllers.

And: "Global optimization methods search the entire domain of admissible solutions. Therefore, they may be unsuitable for learning the control law for cavity-pressure controllers, as poor parameterization can damage the IM machine and lead to scrap parts."

And: "Although the computational effort of MGLBO is substantially higher than that of Vanilla BO, particularly due to model retraining, the optimization is allowed to run for the full cooling time. The cooling time usually makes up more than half of the complete cycle [22]."

And: The proposed approach must be adapted for the industrial application. The transition points between the local expert models are not updated in this work and are assumed to be known.

And: For a real-world application, these points must be updated, by e.g., using the Maximum-Expectation Algorithm [23] or an Interacting Multiple Model filter [24].

The paper also states: Future work will involve testing the algorithm on real IM hardware under realistic conditions and comparing different interpretable control laws with model-based controllers.

And: "Beyond guiding controller optimization, incorporating a process model into the optimization loop enables additional options. One could add process-dependent constraints, such as maximum admissible cavity-pressure overshoots."

And: "Combined with quality-attribute models that use the plant’s simulated pressure trajectories, we could also treat quality attribute references as equality constraints in the controller optimization. This could yield a tightly controlled IMP while meeting desired quality-attribute references."

The paper also states: The control law of a P controller is defined by πP (k) = [θ C]1 · (pref C(k) − pC(k)) and the control law and the internal-state-update equation for the G-PI controller are written as πPI (k) = KPeff e(k) + KIeff cPI (k) and e(k) = pref C(k) − pC(k).

And: The RBF controller exemplifies a highly nonlinear control law with more tunable parameters than the aforementioned controllers. The control law πRC (k) = θ ⊤ Cψ ψi (k) = exp − 1/2 σRC 2 and pC(k) - [r RC]i pref C(k) squared !.

And: The controllers are parameterized such that the initial controller is always a P controller, with gain KPInit.

And: For the GPI controller, enforcing the P control law is straightforward. For the RBF controller, we fitted the weights using least squares and the pseudo-inverse to the P control law.

And: "Given that a negative uS drives a positive QS, the control + signal is constrained to the interval uS ∈ [u−S,lim, uS,lim]. The lower bound addresses the maximum injection speed of the machine, while the upper bound prevents undesirably high screw retraction."

The paper also states: "To assess MGLBO’s performance, we compare it using three metrics. 1) We evaluate the controller performance using the minimum observed cost Jmin. 2) The maximum cost Jmax is used to evaluate the safety of the tuning process. 3) The metric of cumulative worsening JW (n) = n X max(0, Ji − Ji−1) i=2 is also important in industrial applications."

The paper also states: "Although the computational effort of MGLBO is substantially higher than that of Vanilla BO, particularly due to model retraining, the optimization is allowed to run for the full cooling time. The cooling time usually makes up more than half of the complete cycle [22]."

The paper also states: The MGLBO approach demonstrates strong potential for risk-aware, data-efficient controller optimization in the IMP.

Here are the specific improvements that can be made to existing AI systems, based on the proposed Model-Guided Local Bayesian Optimization (MGLBO) framework, and what these improved systems could achieve:

The core improvement is moving from purely data-driven or black-box optimization of control parameters to a hybrid approach that integrates a physics-informed surrogate model with localized, risk-aware Bayesian optimization.

Here are the specific improvements and their resulting capabilities:

1. The system can perform automated controller tuning during manufacturing by optimizing interpretable controllers (P, G-PI, RBF) while simultaneously minimizing cycle time and ensuring risk awareness (mitigating high-cost excursions).

2. The improved system can achieve cost performance comparable to or better than global Bayesian Optimization (BO) over a limited number of iterations, specifically in high-dimensional parameter spaces where Vanilla BO scales poorly.

3. The system can operate efficiently by utilizing a composite objective function that combines a physics-inspired neural network model and a Gaussian Process (GP) regression model to correct the mismatch between simulated costs and real plant observations.

4. The system can identify optimal controller parameters through Local Bayesian Optimization guided by an acquisition function that balances predicted performance against uncertainty, allowing for cautious exploration of the parameter space.

5. The system can provide quantifiable safety metrics, specifically monitoring the maximum cost (risk of machine damage/scrap) and the cumulative worsening of performance, enabling real-time decision-making regarding controller stability.

6. The improved system can be adapted for complex IM scenarios by incorporating process-dependent constraints (e.g., maximum admissible cavity-pressure overshoots) and quality attribute references as equality constraints in the optimization loop, leading to tightly controlled parts that meet specific quality targets while maintaining machine safety.

7. The system can converge more efficiently toward a final, stable controller parametrization compared to Vanilla BO, which often exhibits higher cumulative performance degradation during its search process.

This improved AI system can achieve the following:

1. It will deliver injection molding controllers that are not only highly accurate but also transparent and interpretable (e.g., visualizing the resulting control law as a curve), which is crucial for industrial adoption.

2. It will drastically reduce the time required to find an optimal controller configuration by focusing the search using local trust regions, making it more suitable for rapid manufacturing cycles compared to global optimization methods that search the entire parameter domain.

3. It will significantly enhance operational safety by proactively identifying and avoiding control parameters that lead to dangerous pressure excursions or machine strain (high-cost regions), thereby minimizing scrap rates and physical damage to the machinery.

4. It will provide a robust, data-efficient method for adapting controllers to varying mold geometries and material properties without requiring extensive prior, comprehensive process data.

5. It will enable the development of self-tuning injection molding lines where the control law is automatically optimized in real-time or between cycles based on learned dynamics and operational constraints.

Advanced control methods have proven effective for controlling cavity pressure, a key determinant of part-quality attributes, in the plastics injection molding process. However, the abstract nature of the resulting control laws makes them difficult to interpret in a production environment, thereby limiting adoption in industrial applications. Additionally, controller optimization poses a severe challenge due to the diversity of mold geometries and materials. We propose a method to automatically optimize interpretable controllers during manufacturing while being cycle-efficient and risk-aware. The approach uses a Physics-Inspired Neural Mixture-of-Local-Experts model of the injection molding dynamics and augments its simulated closed-loop costs with a residual Gaussian Process, enabling Local Bayesian Optimization of controller parameters. We benchmark the algorithm against Vanilla Bayesian Optimization (BO) in simulation, using three controllers with parameter counts ranging from 1 to 30. Using the local method, we identify controller parameters that yield costs comparable to or lower than those of global BO over 20 optimization iterations, while mitigating high-cost excursions during tuning.