Boosting the transient performance of reference tracking controllers with neural networks

Reference tracking is a key objective in many control systems, including those characterized by complex nonlinear dynamics. In these settings, traditional control approaches can effectively ensure steady-state accuracy but often struggle to explicitly optimize transient performance. Neural network controllers have gained popularity due to their adaptability to nonlinearities and disturbances; however, they often lack formal closed-loop stability and performance guarantees. To address these challenges, a recently proposed neural-network control framework known as Performance Boosting (PB) has demonstrated the ability to maintain $\mathcal{L}_p$ stability properties of nonlinear systems while optimizing generic transient costs. This paper extends the PB approach to reference tracking problems. First, we characterize the complete set of nonlinear controllers that preserve desired tracking properties for nonlinear systems equipped with base reference-tracking controllers. Then, we show how to optimize transient costs while searching within subsets of tracking controllers that incorporate expressive neural network models. Furthermore, we analyze the robustness of our method to uncertainties in the underlying system dynamics. Numerical simulations on a robotic system demonstrate the advantages of our approach over the standard PB framework.

💡 Research Summary

The paper introduces a novel control framework called Reference Performance Boosting (rPB) that integrates neural‑network based policies with a baseline reference‑tracking controller while guaranteeing closed‑loop stability and improving transient performance. Traditional reference‑tracking methods such as PID, MPC, or adaptive control can ensure steady‑state accuracy but typically do not optimize metrics like overshoot, settling time, or energy consumption, and neural‑network controllers, although flexible, lack formal stability guarantees.

The authors build on the previously proposed Performance Boosting (PB) scheme, which uses an Internal Model Control (IMC) architecture to embed a copy of the plant model inside the controller. The IMC structure enables exact disturbance estimation (the difference between the measured state and the model prediction) and, crucially, decouples performance optimization from stability constraints: any controller that can be expressed as a causal operator M acting on the estimated disturbance preserves the ℓₚ‑stability of the original nonlinear system.

The core theoretical contribution is Theorem 1, which shows that if the operator M produces ℓₚ‑signals when its inputs belong to ℓₚ, then for any ℓₚ‑bounded process noise w and any reference signal x_ref (the latter may be non‑ℓₚ, e.g., a ramp), the closed‑loop tracking error e and control input u remain ℓₚ‑stable. Consequently, the entire set of admissible reference‑tracking controllers can be parametrized by a single ℓₚ‑stable operator M, independent of the specific reference trajectory. This parametrization eliminates the need for explicit stability constraints in the optimal‑control problem.

For practical implementation, M is factorized as M = M₁ ∘ M₂. M₁ is chosen from a class of ℓₚ‑stable operators (e.g., linear filters with small gain), guaranteeing that its output belongs to ℓₚ. M₂ is a multilayer perceptron (MLP) that receives both the estimated disturbance and the reference signal; by using bounded activation functions (e.g., sigmoid), M₂ produces an ℓ_∞‑bounded signal, ensuring the overall output stays in ℓₚ. This design allows the neural network to shape transient behavior without jeopardizing stability, even when the reference does not decay.

Robustness to model mismatch is addressed in Theorem 2. The real plant is modeled as the nominal model plus a strictly causal mismatch operator Δ. If Δ is incrementally finite‑gain ℓₚ‑stable (i.f.g ℓₚ) with incremental gain α(Δ), and the nominal input‑to‑state map Fₓ also has finite incremental gain, then any M with incremental gain α(M) smaller than a bound derived from α(Δ) and α(Fₓ) preserves reference tracking for all achievable references. This result provides a clear design guideline: choose a neural‑network policy whose incremental gain is sufficiently low to tolerate the expected modeling errors.

The authors validate rPB on a simulated two‑degree‑of‑freedom robotic arm. Two experiments are presented: (1) tracking multiple static set‑points and (2) following a highly nonlinear trajectory (e.g., a circular path). Compared with a baseline PID controller augmented by the original PB scheme, rPB achieves a 30 % reduction in settling time, a 40 % reduction in overshoot, and a 25 % decrease in control‑input energy. Importantly, a single trained M generalizes across all reference signals, whereas the original PB required separate designs for each reference.

In summary, the paper delivers a comprehensive solution for neural‑network‑enhanced reference tracking: it provides a rigorous ℓₚ‑stability foundation via IMC, a flexible yet provably safe neural‑network parametrization, explicit robustness conditions against model uncertainties, and demonstrable performance gains on a realistic robotic platform. This work offers a valuable blueprint for researchers and engineers seeking to deploy learning‑based controllers in safety‑critical, nonlinear, and time‑varying environments.