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

TL;DR

The paper tackles control of highly uncertain nonlinear systems by online identification of a discrete-time pseudo-linear input-output (PLIO) model using recursive least squares with subspace information forgetting (RLS-SIFt) and by solving a nonlinear receding-horizon optimization via iterative MPC (IMPC). The approach, termed NPCAC, extends LPCAC by representing nonlinearities with basis functions (polynomial, Fourier, cubic Hermite splines) and updating model parameters online to drive a predictive cost at each step: $\hat{y}_k = \bar{\theta}_k \phi_k$. Across numerical experiments, NPCAC with various bases improves command-following and one-step prediction relative to LPCAC, with cubic Hermite splines (CB$4$) often delivering the strongest performance near the operating region. The results demonstrate the viability of an online, data-free nonlinear MPC framework and identify directions for scaling to higher-order, MIMO, and physically motivated systems, as well as exploring alternative basis families and efficiency gains in NLP solvers.

Abstract

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.

Figures (18)

• Figure 1: Timing diagram for implementing IMPC at step $k$. At step $k$, IMPC uses $y_k$ and $u_k$. Between steps $k$ and $k+1,$$u_k$ is applied and IMPC is executed and computes $u_{k+1}$. Between steps $k+1$ and $k+2$, $u_{k+1}$ is applied. The green-shaded region shows the $\ell$-step prediction horizon.
• Figure 2: Example \ref{['eg1']}: LPCAC of \ref{['eq:atan']}. (a) shows the measurement $y_k$ and the command $r_k$; (b) shows the command-following error ${\rm log}_{10} (| e_{{\rm c},k}|)$; (c) shows the control $u_k$.
• Figure 3: Example \ref{['eg1']}: LPCAC of \ref{['eq:atan']}. (a) shows $F_1$ and its estimate $\widehat{F}_{1,k}$; (b) shows $G_1(y_{k-1})$ and its estimate $\widehat{G}_{1,k}(y_{k-1})$; (c) shows the one-step prediction error ${\rm log}_{10} (| e_{{\rm p},k}|)$.
• Figure 4: Example \ref{['eg3']}: LPCAC of \ref{['eq:atan_1']}. (a) shows the measurement $y_k$ and the command $r_k$; (b) shows the command-following error ${\rm log}_{10} (| e_{{\rm c},k}|)$. (c) shows the control $u_k$. Note that the command-following error is an order of magnitude higher than in Example \ref{['eg1']}.
• Figure 5: Example \ref{['eg3']}: LPCAC of \ref{['eq:atan_1']}. (a) shows $F_1$ and its estimate $\widehat{F}_{1,k}$; (b) shows $G_1(y_{k-1})$ and its estimate $\widehat{G}_{1,k}(y_{k-1})$; (c) shows the one-step prediction error ${\rm log}_{10} (| e_{{\rm p},k}|)$. Note that unlike Figure \ref{['fig:eg1_coef']}(b) in Example \ref{['eg1']}, $G_1(y_{k-1})$ crosses zero.