Risk-Averse Model Predictive Control Framework

Updated 30 June 2025

Risk-averse MPC is an optimization-based feedback strategy that utilizes time-consistent dynamic risk measures to manage uncertainty in multi-stage control problems.
The framework leverages convex optimization with LMI constraints to ensure stability and enable real-time implementation on resource-constrained platforms.
It offers a tunable tradeoff between risk-neutral and robust designs, allowing practitioners to balance nominal performance with enhanced protection against rare events.

Model Predictive Control (MPC) is an optimization-based feedback control strategy in which, at each sampling instant, a finite-horizon optimal control problem is solved for a process model using current state measurements and future predictions. The first control input from the solution is implemented, and the process is repeated at the next time interval with updated measurements. Modern MPC frameworks support a broad class of system dynamics, constraints, and objectives, including uncertainty quantification, risk aversion, and data-driven modeling. Recent advances further leverage formal risk measures, robust optimization, and real-time computational strategies.

1. Time-Consistent Risk-Averse Model Predictive Control

Traditional MPC often optimizes expected cost (risk-neutral) or employs min-max objectives (robust MPC), but these schemes can be either insufficiently protective against rare events or overly conservative. To address this, risk-averse MPC frameworks employ time-consistent dynamic risk metrics as objective functions, particularly for linear systems with multiplicative uncertainty (1511.06981).

Time consistency in risk assessment ensures that risk preferences are rational and do not lead to inconsistent decisions across time steps—a property lacking in naive or static risk-averse MPC designs. This property is critical in multi-stage decision problems to avoid paradoxical outcomes.

The approach is formalized through compositional dynamic risk measures: $\rho_{k,N}(Z_k, \ldots, Z_N) = Z_k + \rho_k \left(Z_{k+1} + \rho_{k+1}(Z_{k+2} + \cdots + \rho_{N-1}(Z_N) \cdots )\right)$ where each $\rho_k$ is a coherent, one-step risk measure, defined axiomatically (convexity, monotonicity, translation invariance, positive homogeneity). These risk measures encompass a spectrum from risk-neutral (expectation) to worst-case, depending on the choice of risk envelope.

The notion of polytopic dual representation for risk measures is operationalized, where the risk metric can be written as a maximum over expectations with respect to alternative weighted probability distributions, whose envelopes are defined as polytopes. This structure enables efficient convex optimization formulations for the MPC law.

2. Algorithmic Structure and Stability

At each time instant, the online risk-averse MPC algorithm proceeds as follows:

State Measurement: Observe the current state $x_k$ .
Risk-Averse Optimization: Solve the following finite-horizon dynamic programming or convex optimization:

$\min_{\pi_{k|k}, \ldots, \pi_{k+N-1|k}} J(x_{k|k}, \pi_{k|k}, \ldots, \pi_{k+N-1|k}, P)$

where

$J(x_{k|k}, \ldots, P) = \rho_{k,k+N}\left(c(x_{k|k}, u_{k|k}), \ldots, c(x_{k+N-1|k}, u_{k+N-1|k}), x_{k+N|k}^T P x_{k+N|k}\right)$

and $c(x, u) = x^T Q x + u^T R u$ .

Actuation: Apply only the first control $u_k$ , and advance to the next time step.
Recursion: Repeat with updated state.

The problem is formulated as a convex quadratic program (QCQP) via the dual polytopic risk structure and includes linear and linear matrix inequality (LMI) constraints to guarantee Uniform Global Risk-Sensitive Exponential Stability (UGRSES). LMIs encode sufficient conditions for stability under the chosen risk metric, extending classical Lyapunov theory to the risk-averse, stochastic setting: $\sum_{j=1}^L q_{l}(j)\, (A_{j}+B_{j}F)^TP(A_j+B_{j}F)-P+(F^TRF+Q)\prec 0$ for all extreme points $q_l$ of the risk envelope.

3. Practical Implementation and Computational Aspects

The risk-averse MPC, by exploiting the Markov dynamic polytopic risk metric, enables the computation of the control law through tractable convex optimization, even under the presence of uncertainty. The optimization can be solved with standard QP or SOCP solvers, and is suitable for embedded, resource-constrained platforms for moderate prediction horizons.

Key design and implementation considerations:

Solving LMIs: LMI constraints for stability are computed offline or online as part of the optimization and are compatible with state-of-the-art convex solvers.
Scenario Trees: The recursive cost requires modeling all possible future uncertainty realizations, which can be efficiently represented through scenario trees for moderate branching factors and horizon lengths.
Control Law Storage: For smaller systems, explicit MPC laws (lookup tables) can be derived using multi-parametric programming, though generally the law is computed online.
Computational Overhead: For the tested 2D system with $N=3$ , the proposed algorithm is tractable and able to meet real-time requirements.

4. Tuning the Risk/Performance Tradeoff

The framework allows direct interpolation between risk-neutral, risk-averse, and robust MPC by adjusting the risk envelope in the dual representation. For example, the mean upper semi-deviation metric with a parameter $c$ provides a continuum from expectation minimization ( $c=0$ ) to worst-case minimization ( $c=1$ ). Increasing risk aversion ( $c\uparrow$ ) yields greater stability and less cost dispersion at the expense of higher average (nominal) cost, enabling practitioners to explicitly balance nominal performance and robustness.

Simulations confirm that as $c$ is increased, average cost rises but the probability of rare, high-cost excursions is sharply curtailed.

5. Comparison with Classical and Robust MPC

Risk-averse MPC, as presented, generalizes both the risk-neutral (standard) and robust (worst-case) MPC frameworks. Unlike robust MPC, which is often overly conservative due to focus on worst-case scenarios, risk-averse MPC provides a principled, quantifiable tradeoff, and avoids time-inconsistency issues found with static risk evaluations. This time-consistent design is critical in applications where risk preferences must not depend paradoxically on time or information.

By choosing different risk envelopes or coherent risk functions (including expectation, CVaR, mean semi-deviation), practitioners can tailor risk sensitivity to application-specific tolerance for adverse events.

6. Theoretical Significance and Impact

The main contributions are:

Axiomatic, time-consistent extension of risk-aversion to MPC, guaranteeing consistent dynamic preferences over multi-stage horizons.
Markov dynamic polytopic risk metrics, which generalize and unify a wide class of risk and robustness concepts in control.
Convex formulation of the finite-horizon risk-averse optimal control problem, ensuring real-time implementation is viable.
Stability guarantees, expressed in the risk-sensitive metric, that extend Lyapunov theory to probabilistic and risk-aware settings.

This advances the practical design of controllers for uncertain systems by enabling consistent, tunable risk management within a tractable optimization-based feedback framework, applicable to a variety of real-world processes where uncertainty and rare events are of concern.

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References (1)

A Framework for Time-Consistent, Risk-Averse Model Predictive Control: Theory and Algorithms (2015)