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Learning Soft Constrained MPC Value Functions: Efficient MPC Design and Implementation providing Stability and Safety Guarantees

Nicolas Chatzikiriakos, Kim P. Wabersich, Felix Berkel, Patricia Pauli, Andrea Iannelli

TL;DR

The paper addresses stability and safety of nonlinear MPC on embedded hardware by learning a continuous value function for a soft constrained MPC, enabling efficient online control. It extends soft constrained MPC to nonlinear systems with constraint tightening, proving local Lipschitz continuity of the value function and input-to-state stability (ISS) of the closed-loop under approximation error. It further proposes a split-value approach to decompose the value function into a smooth performance term and a nonnegative safety term, facilitating supervised learning and ensuring constraint satisfaction. A nonlinear mass-spring-damper example demonstrates feasibility and fast evaluation while preserving the closed-loop behavior and constraint safety.

Abstract

Model Predictive Control (MPC) can be applied to safety-critical control problems, providing closed-loop safety and performance guarantees. Implementation of MPC controllers requires solving an optimization problem at every sampling instant, which is challenging to execute on embedded hardware. To address this challenge, we propose a framework that combines a tightened soft constrained MPC formulation with supervised learning to approximate the MPC value function. This combination enables us to obtain a corresponding optimal control law, which can be implemented efficiently on embedded platforms. The framework ensures stability and constraint satisfaction for various nonlinear systems. While the design effort is similar to that of nominal MPC, the proposed formulation provides input-to-state stability (ISS) with respect to the approximation error of the value function. Furthermore, we prove that the value function corresponding to the soft constrained MPC problem is Lipschitz continuous for Lipschitz continuous systems, even if the optimal control law may be discontinuous. This serves two purposes: First, it allows to relate approximation errors to a sufficiently large constraint tightening to obtain constraint satisfaction guarantees. Second, it paves the way for an efficient supervised learning procedure to obtain a continuous value function approximation. We demonstrate the effectiveness of the method using a nonlinear numerical example.

Learning Soft Constrained MPC Value Functions: Efficient MPC Design and Implementation providing Stability and Safety Guarantees

TL;DR

The paper addresses stability and safety of nonlinear MPC on embedded hardware by learning a continuous value function for a soft constrained MPC, enabling efficient online control. It extends soft constrained MPC to nonlinear systems with constraint tightening, proving local Lipschitz continuity of the value function and input-to-state stability (ISS) of the closed-loop under approximation error. It further proposes a split-value approach to decompose the value function into a smooth performance term and a nonnegative safety term, facilitating supervised learning and ensuring constraint satisfaction. A nonlinear mass-spring-damper example demonstrates feasibility and fast evaluation while preserving the closed-loop behavior and constraint safety.

Abstract

Model Predictive Control (MPC) can be applied to safety-critical control problems, providing closed-loop safety and performance guarantees. Implementation of MPC controllers requires solving an optimization problem at every sampling instant, which is challenging to execute on embedded hardware. To address this challenge, we propose a framework that combines a tightened soft constrained MPC formulation with supervised learning to approximate the MPC value function. This combination enables us to obtain a corresponding optimal control law, which can be implemented efficiently on embedded platforms. The framework ensures stability and constraint satisfaction for various nonlinear systems. While the design effort is similar to that of nominal MPC, the proposed formulation provides input-to-state stability (ISS) with respect to the approximation error of the value function. Furthermore, we prove that the value function corresponding to the soft constrained MPC problem is Lipschitz continuous for Lipschitz continuous systems, even if the optimal control law may be discontinuous. This serves two purposes: First, it allows to relate approximation errors to a sufficiently large constraint tightening to obtain constraint satisfaction guarantees. Second, it paves the way for an efficient supervised learning procedure to obtain a continuous value function approximation. We demonstrate the effectiveness of the method using a nonlinear numerical example.
Paper Structure (15 sections, 6 theorems, 33 equations, 3 figures)

This paper contains 15 sections, 6 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline:
  3. Definitions:
  4. Problem Formulation and MPC Background
  5. Incorporating constraint penalties into value functions using soft constraints
  6. Value function-based approximation of the MPC control law
  7. Stability
  8. Constraint satisfaction
  9. Efficient value function approximation by splitting performance and safety
  10. Numerical Example
  11. Proofs
  12. Proof of Proposition \ref{['prop:SCMPCStability']}
  13. Proof of Theorem \ref{['thm:LipschitzVFcn']}
  14. Proof of Lemma \ref{['lemma:Error_diffU']}
  15. Example of constraint violations

Key Result

Proposition 1

Let Assumption ass:TerminalSetSCMPC hold and consider eq:SCMPC_Problem_full with $\ell (x,u) = \Vert x \Vert_Q^2 + \Vert u \Vert_R^2$. If eq:SCMPC_Problem_full is initially feasible, then it follows under application of $u(k) = \pi_\mathrm{MPC}^\mathrm{s}(x(k))$ that eq:SCMPC_Problem_full is recursi

Figures (3)

  • Figure 1: Visualization of the proof of Theorem \ref{['th:constrSat']}: Optimal value function and corresponding approximation error with appropriate choice of $\rho$ (blue) and $\rho$ selected too small (red). Cases with $\rho$ selected to small can cause constraint violation, see left boundary of $\mathcal{X}$. The level-set $\{x\in\mathbb R^{n_x} | \tilde{V}^\mathrm{s}(x) \le V_\mathrm{max} + 3 \hat{\varepsilon} \}$ bounding closed-loop trajectories has to be contained in $\mathcal{X}$ to guarantee constraint satisfaction.
  • Figure 2: Left: Illustrative value function \ref{['eq:SCMPC_Problem_full']} and its approximation using the split defined in \ref{['eq:value_function_split']} to efficiently parametrize sharply increasing curvatures. Middle/Right: Simulations according to Section \ref{['sec:numerical_example']} under the MPC \ref{['eq:SCMPC_Problem_full']} (solid line) and its approximation \ref{['eq:ApproxControl']} (dashed line), obtained as described in Section \ref{['sec:mpc_value_function_approximation']} using 729 data samples. Middle: Phase trajectories for different initial conditions. Right: Specific state and input trajectory.
  • Figure 3: Trajectories of the closed-loop with *MPC (\ref{['fig:Ex_MPC']}), soft constrained *MPC (\ref{['fig:Ex_SCMPC']}) and approximating control law \ref{['eq:app_ex1_AlterControlLaw']} (\ref{['fig:ExApproxControl']}).

Theorems & Definitions (6)

  • Proposition 1
  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Lemma 5