Risk-averse risk-constrained optimal control

Abstract

Multistage risk-averse optimal control problems with nested conditional risk mappings are gaining popularity in various application domains. Risk-averse formulations interpolate between the classical expectation-based stochastic and minimax optimal control. This way, risk-averse problems aim at hedging against extreme low-probability events without being overly conservative. At the same time, risk-based constraints may be employed either as surrogates for chance (probabilistic) constraints or as a robustification of expectationbased constraints. Such multistage problems, however, have been identified as particularly hard to solve. We propose a decomposition method for such nested problems that allows us to solve them via efficient numerical optimization methods. Alongside, we propose a new form of risk constraints which accounts for the propagation of uncertainty in time. † P. Sopasakis is with Queen's University Belfast,

Learning-Based Distributionally Robust Model Predictive Control of Markovian Switching Systems with Guaranteed Stability and Recursive Feasibility

“…By exploiting the dual risk representation [25, Thm 6.5], the left-hand inequality in ( 9) can be formulated in terms of only linear constraints [9]. As such, it can be used as a tractable surrogate for the original chance constraints, given perfect probabilistic information.…”

“…When represented in this manner, it is clear that ( 13) falls within the class of risk-averse, risk-constrained optimal control problems, described in [9]. In particular, the constraints (13d) at stage k can be represented in the framework of [9] as nested risk constraints which is a composition of a set of conditional risk mappings, in this case consisting of k − max operators and a conditional risk mapping based on (10) at stage k. This is in line with the observations of [28,Sec. 7.1].…”

“…7.1]. Consequently, if the risk measures employed in the definition of the DR-OCP (13) belong to the broad family of conic risk measures, then the reformulations in [9] can be applied to cast (13) as a conic optimization problem which can be effectively solved with available solvers. This is the case for many commonly used coherent risk measures, including the risk measure induced by the 1 -ambiguity set discussed in Example II.4.…”

“…We summarize our contributions as follows. (i) We propose a general formulation for data-driven, distributionally robust model predictive control scheme for Markov switching systems with unknown transition probabilities, which is compatible with the recently developed framework of risk-averse model predictive control [8], [9]. The resulting closed-loop system satisfies the (chance) constraints of the original stochastic problem and allows for online improvement of performance based on observed data.…”

Safe Learning-Based Control of Stochastic Jump Linear Systems: a Distributionally Robust Approach

“…In future work, we aim to generalize this methodology to Markovian disturbances and nonlinear systems. We also aim to study the use of these results to design terminal conditions for risk-averse risk-constrained model predictive control [27]. APPENDIX ♠ Proof of Theorem III.1.…”