An efficient bounded-variable nonlinear least-squares algorithm for embedded MPC

This paper presents a new approach to solve linear and nonlinear model predictive control (MPC) problems that requires small memory footprint and throughput and is particularly suitable when the model and/or controller parameters change at runtime. Typically MPC requires two phases: 1) construct an optimization problem based on the given MPC parameters (prediction model, tuning weights, prediction horizon, and constraints), which results in a quadratic or nonlinear programming problem, and then 2) call an optimization algorithm to solve the resulting problem. In the proposed approach the problem construction step is systematically eliminated, as in the optimization algorithm problem matrices are expressed in terms of abstract functions of the MPC parameters. We present a unifying algorithmic framework based on active-set methods with bounded variables that can cope with linear, nonlinear, and adaptive MPC variants based on a broad class of prediction models and a sum-of-squares cost function. The theoretical and numerical results demonstrate the potential, applicability, and efficiency of the proposed framework for practical real-time embedded MPC.

💡 Research Summary

This paper introduces a novel framework for solving linear and nonlinear Model Predictive Control (MPC) problems on resource‑constrained embedded platforms. Traditional MPC implementations consist of two distinct phases: (i) constructing a quadratic or nonlinear programming problem from the current model, horizon, weighting matrices, and constraints, and (ii) invoking a generic optimizer to solve that problem. When the prediction model or controller parameters change at runtime—common in adaptive, nonlinear, or data‑driven MPC—the construction phase can be as computationally demanding as the solution phase, often precluding real‑time operation.

The authors eliminate the construction phase entirely by expressing all matrices that appear in the optimizer as abstract operators that map the current MPC parameters (model coefficients, weighting matrices, horizon lengths, bounds) directly to the required algebraic results. Consequently, no explicit matrix assembly is required at each sampling instant; the optimizer works with function calls that internally compute the needed products, thereby drastically reducing memory traffic and eliminating the need for dynamic memory allocation.

To handle constraints efficiently, the paper adopts a quadratic penalty formulation for the equality constraints that arise from the prediction model. By adding a term ½ ρ‖h(z,φ)‖² to the cost, the original constrained problem is transformed into a box‑constrained nonlinear least‑squares (NLLS‑box) problem:

 min_{p ≤ z ≤ q} ½‖r(z)‖²,  r(z) =