Nonlinear Predictive Cost Adaptive Control of Pseudo-Linear Input-Output Models Using Polynomial, Fourier, and Cubic Spline Observables

Control of nonlinear systems with high levels of uncertainty is practically relevant and theoretically challenging. This paper presents a numerical investigation of an adaptive nonlinear model predictive control (MPC) technique that relies entirely on online system identification without prior modeling, training, or data collection. In particular, the paper considers predictive cost adaptive control (PCAC), which is an extension of generalized predictive control. Nonlinear PCAC (NPCAC) uses recursive least squares (RLS) with subspace of information forgetting (SIFt) to identify a discrete-time, pseudo-linear, input-output model, which is used with iterative MPC for nonlinear receding-horizon optimization. The performance of NPCAC is illustrated using polynomial, Fourier, and cubic-spline basis functions.

💡 Research Summary

The paper introduces a novel adaptive nonlinear model predictive control (MPC) scheme called Nonlinear Predictive Cost Adaptive Control (NPCAC). The authors start from the well‑established Predictive Cost Adaptive Control (PCAC) framework, which extends Generalized Predictive Control (GPC) by adding a “predictive cost” term that triggers self‑generated persistency of excitation. While PCAC has been shown to work well for many nonlinear plants, it relies on linear input‑output (IO) models identified online with recursive least squares (RLS) and variable‑rate forgetting. Consequently, when the true plant exhibits strong nonlinearities, the linear model can become a poor predictor and the controller’s performance degrades.

To overcome this limitation, the authors propose to model the plant as a discrete‑time pseudo‑linear IO system: the output at time k is expressed as a linear combination of past outputs and inputs, but the coefficients themselves are nonlinear functions of a vector of recent outputs. Formally,

 y_k = Σ_{i=1}^{n}