Optimal shaping filter design for data-driven feedforward controller tuning

Technical Summary: Optimal Shaping Filter Design for Data-Driven Feedforward Controller Tuning

Problem Statement
This paper addresses the data-driven model matching problem within two-degrees-of-freedom (2-DOF) control systems. Specifically, it focuses on tuning a feedforward controller, $C_{ff}(\rho)$, using input-output data from a preliminary experiment to minimize the discrepancy between the closed-loop system, $T(\rho)$, and a desired reference model, $T_d$.

While a common approach involves identifying the "optimal" controller (the one that achieves perfect model matching) directly from data, the paper argues this is insufficient. In scenarios where the controller structure is fixed and cannot perfectly represent the ideal inverse model (i.e., $C^*_{ff} \notin \mathcal{C}$), minimizing the error between the identified controller and the optimal controller does not guarantee the minimization of the actual control performance error. This gap exists because similarity in controller parameters does not immediately imply similarity in closed-loop performance. To bridge this gap, a shaping filter (or prefilter), $L$, is introduced into the cost function. However, designing an optimal shaping filter that relies only on available data, rather than the unknown true plant $P$, remains a significant challenge.

Methodology
The paper formulates the problem as finding a parameter $\rho$ that minimizes the weighted $H_2$ norm of the model matching error:
$$J(\rho) = \|W(T(\rho) - T_d)\|^2$$
where $W$ is a weight function and $P$ is unknown.

The author proposes a data-driven cost function utilizing a shaping filter $L$:
$$J_L(\rho) = \sum_{k=0}^{N-1} \left( L(T_d u^0_k - C_{ff}(\rho) y^0_k) \right)^2$$
where $u^0_k$ and $y^0_k$ are input-output data from a preliminary experiment with an initial feedforward parameter $\rho_0$.

The core methodology involves deriving the optimal $L$ such that minimizing $J_L(\rho)$ is asymptotically equivalent to minimizing the true performance metric $J(\rho)$ as the data length $N \to \infty$. The derivation assumes:

1. The system is noise-free.
2. The initial feedback loop is internally stable.
3. The reference signal spectrum is known (either a step signal generated by a known filter $R$ or white noise filtered by $R$).

Key Contributions
The primary contribution of this work is the explicit derivation of the optimal shaping filter for data-driven feedforward tuning when perfect model matching is unachievable due to structural constraints. The proposed filter is:
$$L = \frac{1}{C_{ff}(\rho_0) + T_d C_{fb}} \frac{W}{R} \frac{1}{z^m}$$
where $C_{fb}$ is the feedback controller, $T_d$ is the reference model, and $z^{-m}$ is a pure delay to ensure the filter is proper.

Key properties of this contribution include:

• Constructibility: The filter is constructed entirely from available information (initial data, known controllers, reference model, and signal spectrum) without requiring knowledge of the unknown plant $P$.
• Optimality: The paper proves that minimizing the cost function with this specific $L$ yields the same optimal parameter $\rho$ as minimizing the true performance cost $J(\rho)$ in the limit of infinite data.
• Relation to ERIT: The paper establishes a theoretical link between the proposed method and Estimated Response Iterative Tuning (ERIT). It demonstrates that ERIT is a special case of the proposed framework and is optimal when the weight function $W$ is chosen to be exactly the reference signal filter $R$ (i.e., when the same reference signal is used in the tuning experiment).

Results
The effectiveness of the proposed shaping filter is validated through two case studies:

1. Numerical Example:

   • A linear time-invariant system with a simple feedforward structure (insufficient to perfectly match the reference model) was used.
   • Case 1 ($W=1$): The optimal shaping filter exhibited high-pass behavior. While the controller tuned without the filter was closer to the "ideal" controller in parameter space, it performed worse in terms of closed-loop tracking. The filter-enabled tuning significantly improved the closed-loop response to match the reference model, particularly in the high-frequency range.
   • Case 2 ($W=R$): This scenario corresponds to the ERIT setup. The results showed that using the shaping filter (equivalent to ERIT) reduced the squared tracking error by approximately 43% compared to tuning without the filter, effectively eliminating oscillations in the output.

2. Practical Experiment:

   • Experiments were conducted on a Quanser Rotary Servo Base Unit with a step reference input.
   • Signal projection was employed to mitigate noise influence.
   • The updated controller using the proposed shaping filter tracked the desired output significantly better than the controller tuned without the filter.
   • The sum of squared errors for the proposed method was 0.26, compared to 0.76 for the method without the shaping filter and 0.81 for the initial controller.

Significance and Claims
The paper claims that the proposed shaping filter provides a rigorous solution to the data-driven model matching problem for feedforward controllers where the controller structure is limited. By constructing the optimal filter from available data, the method avoids the pitfalls of direct controller identification, which can lead to suboptimal performance even with noise-free data.

The author modestly notes that the current work focuses on asymptotic properties (large $N$). They identify the investigation of statistical properties with limited data amounts as a necessary direction for future research. The work reinforces the utility of 2-DOF control structures, where stability is easier to guarantee compared to feedback-only tuning, provided the updated feedforward controller remains stable.