Accelerating soft-constrained MPC for linear systems through online constraint removal

TL;DR

The present paper extends the concept of constraint-adaptive MPC to soft-constrained MPC by detecting and removing constraints based on sub-optimal predicted input sequences, which is rather easy for soft-constrained MPC due to the receding horizon principle and the inclusion of slack variables.

Abstract

Optimization-based controllers, such as Model Predictive Control (MPC), have attracted significant research interest due to their intuitive concept, constraint handling capabilities, and natural application to multi-input multi-output systems. However, the computational complexity of solving a receding horizon problem at each time step remains a challenge for the deployment of MPC. This is particularly the case for systems constrained by many inequalities. Recently, we introduced the concept of constraint-adaptive MPC (ca-MPC) to address this challenge for linear systems with hard constraints. In ca-MPC, at each time step, a subset of the constraints is removed from the optimization problem, thereby accelerating the optimization procedure, while resulting in identical closed-loop behavior. The present paper extends this framework to soft-constrained MPC by detecting and removing constraints based on sub-optimal predicted input sequences, which is rather easy for soft-constrained MPC due to the receding horizon principle and the inclusion of slack variables. We will translate these new ideas explicitly to an offset-free output tracking problem. The effectiveness of these ideas is demonstrated on a two-dimensional thermal transport model, showing a three order of magnitude improvement in online computational time of the MPC scheme.

Figures (5)

• Figure 1: Left: the union between the black and white pixels denote the domain $\Omega$ and the white pixels denote the outputs. Right: the (Gaussian) temperature upper bound.
• Figure 2: The spatially distributed heat loads $\gamma_1,\ \gamma_2$, and $\gamma_3$ (from left to right).
• Figure 3: Top: the temperature at time steps 15, 30, and 45, respectively. Bottom: white pixels indicate that the corresponding constraint is added to $\mathbb{A}$ for at least one step in the horizon. Note that at time step 30, we approximately obtain the maximum number of constraints in the MPC problem.
• Figure 4: The number of constraints in the reduced MPC problem over time. The maximum number of constraints (62) is obtained at time step 27. Recall that the total number of constraints is 2030.
• Figure 5: Computation time comparison between the original MPC setup \ref{['eq:CDC_MPC_condensed']} (), and the ca-MPC setup \ref{['eq:CDC_MPC_condensed_red']} (). The time spent by soft-constrained ca-MPC is broken down in all time requited to compute $\mathbb{A}$ () and solving the resulting quadratic program ().

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3