Towards turnpike-based performance analysis of risk-averse stochastic predictive control

TL;DR

This work extends stochastic MPC to risk-averse settings by employing parameterized risk measures to transform stage costs. A turnpike-based analysis connects abstract MPC on random variables with an implementable, measurement-based MPC, yielding near-optimal averaged performance. The main result shows that with a fixed parameter $\theta$ close to the optimal stationary value $\theta^s$, the averaged closed-loop cost converges to the stationary cost up to a horizon-dependent term $\delta(N)$ and a parameter-mismatch term $\alpha(\|\theta-\theta^s\|)$. This provides a practical pathway to deploy risk-aware MPC in real plants, with performance guarantees and guidance on parameter tuning and horizon choice. Future work includes non-averaged performance, stability analysis, and adaptive online tuning of $\theta$.

Abstract

In this paper, we present performance estimates for stochastic economic MPC schemes with risk-averse cost formulations. For MPC algorithms with costs given by the expectation of stage cost evaluated in random variables, it was recently shown that the guaranteed near-optimal performance of abstract MPC in random variables coincides with its implementable variant coincide using measure path-wise feedback. In general, this property does not extend to costs formulated in terms of risk measures. However, through a turnpike-based analysis, this paper demonstrates that for a particular class of risk measures, this result can still be leveraged to formulate an implementable risk-averse MPC scheme, resulting in near-optimal averaged performance.

Figures (5)

• Figure 1: Optimal state and control trajectories on horizons $N=3,5,7,9,11,13$ for $X_0=1.5$ and $\mathbb{T}[Z] = \rho(Z)$ from equation \ref{['eq:AV@R']} (right) as well as $\mathbb{T}[Z] = \mathbb{E}[Z]$ (left).
• Figure 2: Optimal state and control trajectories on horizons $N=3,5,7,9,11,13$ for $X_0=1.5$ and $\mathbb{T}[Z] = \rho(Z)$ from equation \ref{['eq:risk_KL']} (right) as well as $\mathbb{T}[Z] = \mathbb{E}[Z]$ (left).
• Figure 3: Parameter $\theta$ attaining the optimal cost for the optimal trajectories from Figure \ref{['fig:example2_turnpike']}.
• Figure 4: Optimal stationary cost $\ell(\mathbf{X}^s,\mathbf{U}^s)$ (red dashed) and averaged cost for $\theta^s$ on different horizons $N=6,7,8,9$ (left) and for horizon $N=9$ and different parameters $\theta^s,\theta_1,\theta_2$ (right) for the setting of Example \ref{['ex:example2']}.
• Figure 5: Optimal stationary cost $\ell(\mathbf{X}^s,\mathbf{U}^s)$ (red dashed) and averaged cost for $\theta^s$ on different horizons $N=6,7,8,9$ (left) and for horizon $N=9$ and different parameters $\theta^s,\theta_1,\theta_2$ (right) for the setting of Example \ref{['ex:example1']}.

Theorems & Definitions (16)

• Definition 1: Risk measures
• Remark 2
• Definition 3
• Definition 4: parameterized risk measures
• Example 5: Averaged value-at-risk
• Example 6: $\phi$-divergence risk measures
• Remark 7
• Definition 8: Stationary stochastic processes
• Definition 9: Random variable turnpike
• Definition 10: Types of stochastic turnpike properties