Single-Variable GPC Model Predictive Control Algorithm

Single-variable GPC (Generalized Predictive Control) is a model-based predictive control method capable of effectively handling systems with uncertainty and time-delay characteristics. This algorithm achieves precise control of single-variable systems through online rolling optimization combined with feedback correction mechanisms. The implementation typically involves constructing a CARIMA (Controlled Auto-Regressive Integrated Moving Average) model and solving Diophantine equations for multi-step output prediction.

The core concept of GPC utilizes historical data and system models to predict output values for multiple future steps, then calculates optimal control inputs by optimizing an objective function. In single-variable systems, GPC stability depends on the selection of prediction horizon, control horizon, and weighting parameters. Proper parameter configuration ensures system robustness, while overly aggressive control strategies may cause oscillations or instability. Key implementation steps include: designing a cost function with output tracking error and control effort terms, solving the optimization problem using recursive least squares or quadratic programming methods, and implementing receding horizon control with each sampling period.

When studying GPC stability, analysis typically involves examining the closed-loop system's characteristic equation or employing methods like Lyapunov functions. Additionally, performance verification through simulation examples enables直观 observation of system response characteristics under different parameters. Compared to traditional PID control, GPC demonstrates superior performance in handling nonlinear and time-varying systems, though it involves higher computational complexity. MATLAB implementation often uses functions like 'idpoly' for system identification and custom optimization routines for solving the predictive control law.