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Probabilistically Input-to-State Stable Stochastic Model Predictive Control

Maik Pfefferkorn, Rolf Findeisen

TL;DR

The concept of input-to-state stability in probability is exploited and it is outlined how it can be used to provide stability guarantees, circumventing the requirement for strict repeated feasibility guarantees.

Abstract

Employing model predictive control to systems with unbounded, stochastic disturbances poses the challenge of guaranteeing safety, i.e., repeated feasibility and stability of the closed-loop system. Especially, there are no strict repeated feasibility guarantees for standard stochastic MPC formulations. Thus, traditional stability proofs are not straightforwardly applicable. We exploit the concept of input-to-state stability in probability and outline how it can be used to provide stability guarantees, circumventing the requirement for strict repeated feasibility guarantees. Loss of feasibility is captured by a back-up controller, which is explicitly taken into account in the stability analysis. We illustrate our findings using a numeric example.

Probabilistically Input-to-State Stable Stochastic Model Predictive Control

TL;DR

The concept of input-to-state stability in probability is exploited and it is outlined how it can be used to provide stability guarantees, circumventing the requirement for strict repeated feasibility guarantees.

Abstract

Employing model predictive control to systems with unbounded, stochastic disturbances poses the challenge of guaranteeing safety, i.e., repeated feasibility and stability of the closed-loop system. Especially, there are no strict repeated feasibility guarantees for standard stochastic MPC formulations. Thus, traditional stability proofs are not straightforwardly applicable. We exploit the concept of input-to-state stability in probability and outline how it can be used to provide stability guarantees, circumventing the requirement for strict repeated feasibility guarantees. Loss of feasibility is captured by a back-up controller, which is explicitly taken into account in the stability analysis. We illustrate our findings using a numeric example.

Paper Structure

This paper contains 13 sections, 6 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stochastic Model Predictive Control
  3. Stochastic Model Predictive Control Formulation
  4. Remarks on Feasibility and Stability of SMPC
  5. Probabilistically Input-to-State Stable SMPC
  6. Input-to-State Stability in Probability
  7. ISSp under the Back-Up Controller
  8. ISSp under the Stochastic MPC
  9. ISSp under the Combined Control Policy
  10. Numeric Example
  11. ISSp of Autonomous Dynamics
  12. ISSp in Closed-Loop
  13. Conclusions

Key Result

Proposition 1

(From Culbertson2023). If there exists an ISSp Lyapunov function for $x_{k+1} = f(x_k, w_k)$ on $\Omega$, then $x_{k+1} = f(x_k, w_k)$ is ISSp on $\Omega$.

Figures (3)

  • Figure 1: State trajectories of the uncertain autonomous system. Left column: 100 sample trajectories starting at the same initial state. Right column: 100 sample trajectories starting at different initial states.
  • Figure 2: Evolution of \ref{['eq:sim_example_lyapunov']} along the 100 state trajectories of the uncertain autonomous system shown in Figure \ref{['fig:autonomous_system__state']}, left column.
  • Figure 3: Top: State trajectories of system \ref{['eq:example_system']} in closed-loop over time. Bottom: Corresponding phase plane plot. The gray-shaded are indicates the state constraints, while the green-shaded area is the terminal region of MPC \ref{['eq:smpc']}.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 1
  • Theorem 1
  • proof
  • ...and 8 more