MPC-guided, Data-driven Fuzzy Controller Synthesis

TL;DR

This work tackles the challenge of implementing MPC in resource-constrained settings by learning a low-complexity, data-driven surrogate. It trains multiple ARMA controllers on MPC closed-loop data and blends them with a Takagi-Sugeno fuzzy interpolator to form the F-ARMA controller, which operates using measured inputs $y_k$ and references $r_k$ without requiring full state estimation. The approach combines LS/regularized regression to synthesize ARMA coefficients with QP-based training under input constraints, enabling fast online computation while preserving MPC-like response. Numerical examples show that F-ARMA can closely mimic MPC trajectories with orders of magnitude faster execution, highlighting practical potential for embedded or real-time control applications.

Abstract

Model predictive control (MPC) is a powerful control technique for online optimization using system model-based predictions over a finite time horizon. However, the computational cost MPC requires can be prohibitive in resource-constrained computer systems. This paper presents a fuzzy controller synthesis framework guided by MPC. In the proposed framework, training data is obtained from MPC closed-loop simulations and is used to optimize a low computational complexity controller to emulate the response of MPC. In particular, autoregressive moving average (ARMA) controllers are trained using data obtained from MPC closed-loop simulations, such that each ARMA controller emulates the response of the MPC controller under particular desired conditions. Using a Takagi-Sugeno (T-S) fuzzy system, the responses of all the trained ARMA controllers are then weighted depending on the measured system conditions, resulting in the Fuzzy-Autoregressive Moving Average (F-ARMA) controller. The effectiveness of the trained F-ARMA controllers is illustrated via numerical examples.

Figures (8)

• Figure 1: Sampled-data implementation of discrete-time, reference tracking controller applied to a continuous-time system ${\mathcal{M}}$ with input $u$ and output $y.$ All sample-and-hold operations are synchronous.
• Figure 2: Example \ref{['ex:double_integrator']}: Double Integrator. Results from implementing linear MPC on the double integrator dynamics given by \ref{['eq:xeq_ex1']},\ref{['eq:yeq_ex1']} for $x(0) = 00^{\rm T}$. Controller coefficient vector $\theta_1$ corresponding to the first F-ARMA linear controller is trained using the $y$ and $u$ data corresponding to $t \in [0, 6]$ s, that is, all the data shown in the figure. Controller coefficient vector $\theta_2$ corresponding to the second F-ARMA linear controller is trained using the $y$ and $u$ data corresponding to $t \in [1.5, 6]$ s, that is, all the data from $t = 1.5$ s, denoted by the vertical, dashed green line, onwards.
• Figure 3: Example \ref{['ex:double_integrator']}: Double Integrator. Fuzzy membership functions $\mu_1$ and $\mu_2$ corresponding to the fuzzy sets in which $\gamma_k$ is large and small, respectively.
• Figure 4: Example \ref{['ex:double_integrator']}: Double Integrator. Results from implementing the trained F-ARMA controller on the double integrator dynamics given by \ref{['eq:xeq_ex1']},\ref{['eq:yeq_ex1']} for $x(0) = 00^{\rm T}$. The response of F-ARMA is compared against the response of linear MPC and the individually trained ARMA controllers with $\theta_1$ and $\theta_2$ implemented using \ref{['eq:ARMA']}.
• Figure 5: Example \ref{['ex:cart_pendulum']}: Inverted Pendulum on Cart. The pendulum ${\mathcal{P}}$ is attached to the cart ${\mathcal{C}}$ at the pivot point $c,$$p$ is the horizontal position from $c$ to a reference point in the ground $w,$$\phi$ is the angle of ${\mathcal{P}}$ relative to its upwards position, and the horizontal force $F$ is applied to ${\mathcal{C}}.$ The acceleration due to gravity $g$ has a downwards direction.

Theorems & Definitions (2)

• Example VI.1
• Example VI.2