NMPC

Nonlinear model predictive control (NMPC) is a control strategy that uses a nonlinear model of a system to predict its future behavior and optimize control actions over a finite horizon. At each control step, NMPC solves an online optimization problem that minimizes a performance index subject to the system dynamics and constraints, then implements the first control input from the optimal sequence and repeats the process in a receding horizon fashion.

Formulation typically involves a nonlinear dynamic model x_{k+1} = f(x_k, u_k) with state x and input u, possibly outputs y = h(x_k, u_k). The objective is to minimize J = sum_{i=0}^{N-1} L(x_i, u_i) + F(x_N), subject to the dynamics x_{i+1} = f(x_i, u_i) and any inequality constraints g(x_i, u_i) ≤ 0, along with optional terminal constraints or a terminal cost. The optimization is performed online, and only the first control input u_0 is applied before re-solving at the next step.

Key features and considerations include handling nonlinear dynamics and hard constraints directly, and the need to solve a nonlinear programming problem in real time. Computational burden is a major concern, growing with model complexity, horizon length, and nonlinearity. Practitioners address this with fast solvers, warm-starting, model simplifications or reduced-order models, and, in some cases, explicit NMPC where control laws are precomputed offline.

Applications span process control, chemical and energy systems, automotive and robotic applications, aerospace, and other domains where nonlinear behavior and constraints are important. NMPC offers improved performance and constraint satisfaction when nonlinearities are significant, but it requires careful attention to model accuracy, computation time, and robustness.