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Measured-state conditioned recursive feasibility for stochastic model predictive control

Mirko Fiacchini, Martina Mammarella, Fabrizio Dabbene

TL;DR

A stochastic MPC scheme is constructed, based on the introduction of ellipsoidal probabilistic reachable sets, which implements a closed-loop initialization strategy, and is proven to satisfy the novel definition of recursive feasibility.

Abstract

In this paper, we address the problem of designing stochastic model predictive control (MPC) schemes for linear systems affected by unbounded disturbances. The contribution of the paper is twofold. First, motivated by the difficulty of guaranteeing recursive feasibility in this framework, due to the nonzero probability of violating chance-constraints in the case of unbounded noise, we introduce the novel definition of measured-state conditioned recursive feasibility in expectation. Second, we construct a stochastic MPC scheme, based on the introduction of ellipsoidal probabilistic reachable sets, which implements a closed-loop initialization strategy, i.e., the current measured-state is employed for initializing the optimization problem. This new scheme is proven to satisfy the novel definition of recursive feasibility, and its superiority with respect to open-loop initialization schemes, arising from the fact that one never neglects the information brought by the current measurement, is shown through numerical examples.

Measured-state conditioned recursive feasibility for stochastic model predictive control

TL;DR

A stochastic MPC scheme is constructed, based on the introduction of ellipsoidal probabilistic reachable sets, which implements a closed-loop initialization strategy, and is proven to satisfy the novel definition of recursive feasibility.

Abstract

In this paper, we address the problem of designing stochastic model predictive control (MPC) schemes for linear systems affected by unbounded disturbances. The contribution of the paper is twofold. First, motivated by the difficulty of guaranteeing recursive feasibility in this framework, due to the nonzero probability of violating chance-constraints in the case of unbounded noise, we introduce the novel definition of measured-state conditioned recursive feasibility in expectation. Second, we construct a stochastic MPC scheme, based on the introduction of ellipsoidal probabilistic reachable sets, which implements a closed-loop initialization strategy, i.e., the current measured-state is employed for initializing the optimization problem. This new scheme is proven to satisfy the novel definition of recursive feasibility, and its superiority with respect to open-loop initialization schemes, arising from the fact that one never neglects the information brought by the current measurement, is shown through numerical examples.
Paper Structure (12 sections, 27 equations, 5 figures, 1 table)

This paper contains 12 sections, 27 equations, 5 figures, 1 table.

Table of Contents

  1. Probability bounds comparison
  2. Comparison of IS-SMPC and MS-SMPC
  3. Conclusion and future works
  4. Proof of Theorem \ref{['th:rec_feas']}
  5. Proof of Theorem \ref{['th:feasibility']}
  6. Proof of condition (i)
  7. Proof of condition (ii)
  8. Proof of \ref{['eq:condru']}
  9. Proof of \ref{['eq:condrx']}
  10. Proof of \ref{['eq:condr']}
  11. Proof of condition (iii)
  12. Proof of Theorem \ref{['th:balanced_cost']}

Figures (5)

  • Figure 2: State and input trajectories for SMPC enforcing no bounds (solid lines), hard bounds (dotted lines), and soft bounds (dashed lines) on the first input $v_{0|k}$.
  • Figure 3: Comparison among the realized state trajectories starting from $x_0=(-40,\,40)$ for $k \in [0,10]$ enforcing on $v_{0|k}$: a) no bounds (solid line); b) hard bounds (dotted line); and, c) soft bounds (dashed lined).
  • Figure 4: Cost comparison over 1000 simulations with $N=10$ for $x_0 = (-40,\, 40)$ of the three input bounds strategies. Mean cost: no bounds $J_{MPC} = 10006$; hard bounds $J_{MPC}=15467$; soft bounds $J_{MPC}=12173$.
  • Figure 5: Cost comparison over 1000 simulations with $N=10$ for $x(0) = (-30, 0)$. Mean cost: for the new MS-SMPC (with no initial bound) $J_{MPC}=2952$, for the IS-SMPC $J_{MPC}=2953$, with almost unitary ratio.
  • Figure 6: Cost comparison over 1000 simulations of 10 steps for $x(0) = (-40, 37)$, close to the IS-SMPC feasibility bounds. Mean cost: for the MS-SMPC (with no initial bound) $8593$, for the IF-SMPC $11105$, with ratio of around $0.77$.

Theorems & Definitions (2)

  • proof
  • proof