Iterative Learning Control (ILC)

ILC fundamentally requires a finite-duration, repetitive task. The system must reset to identical initial conditions before each trial. This is distinct from adaptive control, which handles continuously varying references. In DPD, this maps perfectly to the transmission of standardized communication frames where the preamble and pilot sequences are known and repeatable.

Unlike feedback controllers that react to current errors, ILC computes a feedforward signal for the next iteration. The learning law updates the input trajectory based on the error trajectory from the previous run:

• P-type ILC: Updates proportional to the previous error.
• D-type ILC: Updates proportional to the error derivative.
• Q-filter: A low-pass filter applied to the error to ensure monotonic convergence and reject high-frequency noise that could destabilize the learning process.

A critical distinction in ILC design. Asymptotic stability guarantees the error goes to zero as iterations approach infinity, but the error may grow catastrophically large in early iterations. Monotonic convergence ensures the error norm strictly decreases at every iteration. This is enforced by designing the learning gain such that the induced operator norm is less than one, a condition often analyzed in the frequency domain.

The foundational D-type learning law proposed by Suguru Arimoto in 1984. The update law is: u_{k+1}(t) = u_k(t) + Γ * d/dt[e_k(t)] where Γ is the learning gain matrix. This formulation proved that a robot manipulator could achieve perfect tracking without an accurate dynamic model, relying solely on the derivative of the tracking error from the previous trial.

A model-based optimization framework that computes the input update by minimizing a quadratic cost function: J = ||e_{k+1}||^2_Q + ||u_{k+1} - u_k||^2_R The R matrix penalizes aggressive input changes, while Q penalizes tracking error. This framework naturally handles non-minimum phase systems and allows explicit tuning of the robustness-to-convergence-speed tradeoff, making it suitable for precision DPD coefficient refinement.

In DPD systems, ILC refines the predistorter coefficients across repeated transmission frames. The process:

• Trial k: Transmit a frame, capture the PA output, compute the error between desired linear output and actual output.
• Update: Adjust DPD coefficients using the error trajectory.
• Trial k+1: Transmit with updated predistorter. This bypasses the model extraction step entirely, directly tuning the predistorter to minimize the observed nonlinear distortion without requiring an explicit PA behavioral model.