Dampening parameter distributional shifts under robust control and gain scheduling

Many traditional robust control approaches assume linearity of the system and independence between the system state-input and the parameters of its approximant (possibly lower-order) model. This assumption implies that the application of robust control design to the underlying system introduces no distributional shifts in the parameters of its approximant model. This is generally not true when the underlying system is nonlinear, which may require different approximant models with different parameter distributions when operated at different regions of the state-input space. Therefore, a robust controller has to be robust under the approximant model with parameter distribution that will be experienced in the future data, after applying this control, not the parameter distribution seen in the learning data or assumed in the design. In this paper, we seek a solution to this problem by restricting the newly designed closed-loop system to be consistent with the learning data and slowing down any distributional shifts in the state-input space of the underlying system, and therefore, in the parameter space of its approximant model. In computational terms, the objective of dampening the shifts in the parameter distribution is formulated as a convex semi-definite program that can be solved efficiently by standard software packages. We evaluate the proposed approach on a simple yet telling gain-scheduling problem, which can be equivalently posed as a robust control problem.

💡 Research Summary

The paper addresses a fundamental limitation of traditional robust control and gain‑scheduling methods: they assume a linear plant and a fixed, state‑input‑independent parameter distribution for the low‑order approximant model. In nonlinear systems, applying a new control law changes the state‑input distribution, which in turn shifts the distribution of the model parameters that the controller relies on. This shift can invalidate the quadratic‑stability condition that underpins robust control guarantees, leading to poor performance or even instability.

To mitigate this problem, the authors introduce a “data‑conforming” framework. The central idea is to constrain the closed‑loop system so that its state‑input distribution after control implementation remains close to the distribution observed in the learning data (or the grid used for gain‑scheduling). They quantify the distance between the design covariance Γ_des and the data covariance Γ_data using Jeffreys divergence, a symmetric form of the Kullback‑Leibler divergence that is convex and has a unique minimum when the two covariances coincide.

Mathematically, the plant is modeled by a difference‑inclusion
 x_{k+1}=F_k x_k+G_k u_k, (F_k,G_k)∈C=conv{(A_i,B_i)}_{i=1}^n,
which can be obtained either from data (uncertainty set) or from Jacobians evaluated on a grid (gain‑scheduling). The standard robust LQR problem is expressed as a semi‑definite program (SDP) with decision variables Σ (state covariance), L=KΣ, and Z₀ (upper bound on KΣKᵀ). The quadratic‑stability condition translates into a set of linear matrix inequalities (LMIs).

The data‑conforming extension adds four auxiliary variables Z₁,…,Z₄ and three new LMIs:

1. Z₁ ≽ V ⎡ Σ L ⎤ ⎣ Lᵀ Σ ⎦, which forces the design covariance to dominate the term V⁻¹+Σ⁻¹KᵀK.
2. Z₃ I ≽ Σ_data, ensuring that the design covariance is not smaller than the empirical covariance from the learning set.
3. A linear matrix inequality involving Z₂, H_data, and Σ⁻¹ that approximates the second term of Jeffreys divergence (Γ_des⁻¹).

These constraints embed the Jeffreys‑divergence regularization directly into the SDP objective:
 min  tr(QΣ)+tr(RZ₀) + γ tr(Γ_data⁻¹ Z₁) + tr(V⁻¹ Z₂) + tr(Σ_data Z₃).

The resulting problem (13) remains a convex SDP, solvable with standard tools such as CVX or MOSEK, and retains the computational scalability of classic robust LQR formulations.

Two theoretical results are proved. First, the optimal Σ* from the data‑conforming SDP is an upper bound on the true steady‑state covariances Σ_i associated with each vertex (A_i,B_i) under the optimal gain K*. Consequently, the actual state‑input distribution is guaranteed to lie inside the design distribution (Γ_true ≼ Γ_des). Second, feasibility of the original robust LQR SDP implies feasibility of the data‑conforming SDP, showing that the added regularization does not over‑constrain the problem.

A numerical illustration on a simple nonlinear system demonstrates that the conventional robust LQR enlarges the uncertainty set dramatically after control implementation, violating quadratic stability, whereas the data‑conforming approach keeps Γ_des close to Γ_data, preserving stability and achieving lower cost.

In summary, the paper makes three key contributions: (1) it highlights how control‑induced distributional shifts can invalidate robust control assumptions; (2) it integrates a statistically grounded regularization (Jeffreys divergence) into the robust control design via LMIs, preserving the convex SDP structure; and (3) it provides theoretical guarantees of feasibility and conservatism, together with empirical evidence that the method yields more reliable performance on nonlinear plants. This work bridges data‑driven system identification and classical robust control, offering a practical pathway to robustly control nonlinear systems while respecting the statistical properties of the available data.