Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularized Model Predictive Control

Komeil Nosrati, Juri Belikov, Aleksei Tepljakov, Eduard Petlenkov

TL;DR

This paper addresses fixed cost matrices in model predictive control by introducing Regularized MPC (Re-MPC), which adaptively updates the design (Hessian) matrix through a recursive PLS-based Riccati equation. A penalized least-squares formulation with a large penalty $\mu$ converts the constrained horizon problem into an unconstrained one, yielding an optimal recursive input $\bar{U}_k^*=K_k x_{k|k}$ and a Riccati recursion for $P_{k+l-1}$ that updates the design matrix at each forward shift. Under controllability and detectability, the authors prove the existence of a unique positive definite solution $P_l\succ0$ for all horizons and establish stability of the closed-loop as the horizon shifts, with feasibility maintained via $fmincon$ and $\mu\to\infty$ driving the residual term to zero. Numerical simulations show that Re-MPC achieves up to about 15% better regulation performance and lower total cost than conventional MPC, while providing a principled, online-tunable design-matrix update with potential extensions to nonlinear and robust MPC.

Abstract

In model predictive control (MPC), the choice of cost-weighting matrices and designing the Hessian matrix directly affects the trade-off between rapid state regulation and minimizing the control effort. However, traditional MPC in quadratic programming relies on fixed design matrices across the entire horizon, which can lead to suboptimal performance. This letter presents a Riccati equation-based method for adjusting the design matrix within the MPC framework, which enhances real-time performance. We employ a penalized least-squares (PLS) approach to derive a quadratic cost function for a discrete-time linear system over a finite prediction horizon. Using the method of weighting and enforcing the constraint equation by introducing a large penalty parameter, we solve the constrained optimization problem and generate control inputs for forward-shifted horizons. This process yields a recursive PLS-based Riccati equation that updates the design matrix as a regularization term in each shift, forming the foundation of the regularized MPC (Re-MPC) algorithm. To accomplish this, we provide a convergence and stability analysis of the developed algorithm. Numerical analysis demonstrates its superiority over traditional methods by allowing Riccati equation-based adjustments.

Regularized Model Predictive Control

TL;DR

This paper addresses fixed cost matrices in model predictive control by introducing Regularized MPC (Re-MPC), which adaptively updates the design (Hessian) matrix through a recursive PLS-based Riccati equation. A penalized least-squares formulation with a large penalty converts the constrained horizon problem into an unconstrained one, yielding an optimal recursive input and a Riccati recursion for that updates the design matrix at each forward shift. Under controllability and detectability, the authors prove the existence of a unique positive definite solution for all horizons and establish stability of the closed-loop as the horizon shifts, with feasibility maintained via and driving the residual term to zero. Numerical simulations show that Re-MPC achieves up to about 15% better regulation performance and lower total cost than conventional MPC, while providing a principled, online-tunable design-matrix update with potential extensions to nonlinear and robust MPC.

Abstract

In model predictive control (MPC), the choice of cost-weighting matrices and designing the Hessian matrix directly affects the trade-off between rapid state regulation and minimizing the control effort. However, traditional MPC in quadratic programming relies on fixed design matrices across the entire horizon, which can lead to suboptimal performance. This letter presents a Riccati equation-based method for adjusting the design matrix within the MPC framework, which enhances real-time performance. We employ a penalized least-squares (PLS) approach to derive a quadratic cost function for a discrete-time linear system over a finite prediction horizon. Using the method of weighting and enforcing the constraint equation by introducing a large penalty parameter, we solve the constrained optimization problem and generate control inputs for forward-shifted horizons. This process yields a recursive PLS-based Riccati equation that updates the design matrix as a regularization term in each shift, forming the foundation of the regularized MPC (Re-MPC) algorithm. To accomplish this, we provide a convergence and stability analysis of the developed algorithm. Numerical analysis demonstrates its superiority over traditional methods by allowing Riccati equation-based adjustments.
Paper Structure (6 sections, 5 theorems, 39 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 6 sections, 5 theorems, 39 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Re-MPC Algorithm
  4. Convergence and Stability of Re-MPC
  5. Simulation Study
  6. Conclusions

Key Result

Lemma 1

Let the matrix $G$ have full column rank, i.e., $\mathop{\mathrm{rank}}\nolimits (G) = m$, and define $\Theta_1 = G^\mathsf{T} W G$ and Then, $\Theta_1$ and $\Theta_2$ are invertible matrices. This guarantees that the optimization problem eq6 has a unique optimal solution, given byIn the case that $W \succeq 0$, the solution is $\hat{\eta} = (G^\mathsf{T} W G)^{\dagger} G^\mathsf{T} W h$. Thus, f

Figures (2)

  • Figure 1: Stabilized states and optimal control effort for C-MPC and Re-MPC.
  • Figure 2: Stabilized state and control effort for varying $\mu$.

Theorems & Definitions (14)

  • Lemma 1: ref20
  • Lemma 2: ref21
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof
  • ...and 4 more