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Model Predictive Control for setpoint tracking

Daniel Limon, Antonio Ferramosca, Ignacio Alvarado, Teodoro Alamo

TL;DR

This paper introduces MPCT, an MPC variant for tracking that augments the controller with an artificial steady state and an offset term to handle changes in the setpoint without sacrificing recursive feasibility. By formulating both equality- and inequality-terminal variants, it proves asymptotic convergence to reachable setpoints and, when necessary, to the closest feasible equilibrium, while maintaining constraint satisfaction under polyhedral constraints. The approach yields convex QP problems, with a path to explicit, off-line control laws via multiparametric QP, and demonstrates larger domains of attraction than standard regulation MPC through a derived invariant terminal set. Practical implications include robust tracking in constrained systems, improved handling of unreachable setpoints, and potential for rapid deployment in industry via explicit MPCT controllers. The four-tanks example illustrates the method's ability to track reachable references, minimize offset for unreachable references, and achieve local optimality under appropriate offset penalties.

Abstract

The main objective of tracking control is to steer the tracking error, that is the difference between the reference and the output, to zero while the plant's operation limits are satisfied. This requires that some assumptions on the evolution of the future values of the reference must be taken into account. Typically a simple evolution of the reference is considered, such as step, ramp, or parabolic reference signals. It is important to notice that the tracking problem considers possible variations in the reference to be tracked, such as steps or slope variations of the ramps. Then the tracking control problem is inherently uncertain, since the reference may differ from what is expected. If the value of the reference is changed, then there is no guarantee that the feasibility and stability properties of the resulting control law hold. This report presents the MPC for tracking (MPCT) approach, which ensures recursive feasibility and asymptotic stability of the setpoint when the value of the reference is changed.

Model Predictive Control for setpoint tracking

TL;DR

This paper introduces MPCT, an MPC variant for tracking that augments the controller with an artificial steady state and an offset term to handle changes in the setpoint without sacrificing recursive feasibility. By formulating both equality- and inequality-terminal variants, it proves asymptotic convergence to reachable setpoints and, when necessary, to the closest feasible equilibrium, while maintaining constraint satisfaction under polyhedral constraints. The approach yields convex QP problems, with a path to explicit, off-line control laws via multiparametric QP, and demonstrates larger domains of attraction than standard regulation MPC through a derived invariant terminal set. Practical implications include robust tracking in constrained systems, improved handling of unreachable setpoints, and potential for rapid deployment in industry via explicit MPCT controllers. The four-tanks example illustrates the method's ability to track reachable references, minimize offset for unreachable references, and achieve local optimality under appropriate offset penalties.

Abstract

The main objective of tracking control is to steer the tracking error, that is the difference between the reference and the output, to zero while the plant's operation limits are satisfied. This requires that some assumptions on the evolution of the future values of the reference must be taken into account. Typically a simple evolution of the reference is considered, such as step, ramp, or parabolic reference signals. It is important to notice that the tracking problem considers possible variations in the reference to be tracked, such as steps or slope variations of the ramps. Then the tracking control problem is inherently uncertain, since the reference may differ from what is expected. If the value of the reference is changed, then there is no guarantee that the feasibility and stability properties of the resulting control law hold. This report presents the MPC for tracking (MPCT) approach, which ensures recursive feasibility and asymptotic stability of the setpoint when the value of the reference is changed.
Paper Structure (21 sections, 8 theorems, 109 equations, 15 figures)

This paper contains 21 sections, 8 theorems, 109 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Problem Description
  3. The set of reachable setpoints
  4. MPC for tracking formulation
  5. MPC for tracking with terminal equality constraint
  6. Stabilizing design
  7. MPCT with terminal inequality constraint
  8. Stabilizing design
  9. Calculation of the invariant set for tracking
  10. Implementation: how to formulate the QP problem
  11. MPCT with terminal inequality constraint
  12. MPCT with terminal equality constraint
  13. Off-line implementation as an explicit controller.
  14. Properties of the proposed controller
  15. Local Optimality
  16. ...and 6 more sections

Key Result

Theorem 1

Consider that Assumptions assumption1_OPT, chaMPCT:def_optimo and chaMPCT:assumptionQR_eqconst hold and the prediction horizon is such that $N\geq n_c$. Then for a given setpoint $y_{sp}$ and for any feasible initial state $x_0 \in \mathcal{X}_N$, the system controlled by the MPC controller $\kappa_

Figures (15)

  • Figure 1: Feasibility: the initial condition $x(0)$ is clearly infeasible for an MPC for tracking $x_{sp2}$ with terminal constraint $x(N)=x_{sp2}$. However, it would be feasible for an MPC for tracking $x_a$ with terminal constraint $x(N)=x_a$.
  • Figure 2: Feasibility: the offset cost function forces $x_a$ to move toward $x_{sp2}$ in order to ensure convergence. The MPC optimization problem remain feasible from the new initial condition $x(0)$.
  • Figure 3: State space evolution of the first 5 steps of simulation: the artificial reference $x_a$ (dahsed-dotted line) never leaves $\mathcal{X}_s$ (dashed line) and moves toward $x_{sp}$ (star).
  • Figure 4: State space evolution of the complete simulation: both artificial reference (dash-dotted line) and closed-loop system (solid line) converge to the desired setpoint (star).
  • Figure 5: Time evolution of the complete simulation: the artificial reference (dashed line) converges to the desired setpoint (dash-dotted line). The closed-loop system (solid line) follows the artificial reference and eventually converges to $y_{sp}$.
  • ...and 10 more figures

Theorems & Definitions (12)

  • Example 1
  • Example 2
  • Theorem 1: Asymptotic Stability
  • Theorem 2: Stability
  • Example 3
  • Lemma 1: Local Optimality
  • Example 4
  • Lemma 2
  • Theorem 3: Region of local optimality
  • Lemma 3
  • ...and 2 more