Abstract
Stochastic linear quadratic control problems are considered from the viewpoint of risks. In particular, a worst-case conditional value-at-risk (CVaR) of quadratic objective function is minimized subject to additive disturbances whose first two moments of the distribution are known. The study focuses on three problems of finding the optimal feedback gain that minimizes the quadratic cost of: stationary distribution, one-step, and infinite time horizon. For the stationary distribution problem, it is proved that the optimal control gain that minimizes the worst-case CVaR of the quadratic cost is equivalent to that of the standard (stochastic) linear quadratic regulator. For the one-step problem, an approach to an optimal solution as well as analytical suboptimal solutions are presented. For the infinite time horizon problem, two suboptimal solutions that bound the optimal solution and an approach to an optimal solution for a special case are discussed. The presented theorems are illustrated with numerical examples.
Highlights
R ISK has been extensively studied in the financial industry [1]– [4], but it is not the only field that requires consideration of risk
One of the measures of risk is Conditional Value-at-Risk (CVaR), which is defined as the conditional expectation of losses exceeding a threshold
Since CVaR provides a convex conservative approximation for a joint chance constraint [17], and enjoys nice mathematical properties of a coherent risk measure [18]–[20], it has been used in portfolio optimizations [1], [2]
Summary
R ISK has been extensively studied in the financial industry [1]– [4], but it is not the only field that requires consideration of risk. Only partial information on the underlying probability distribution is known in many situations This may be because uncertainties come in many different ways and difficult to model. We derive the optimal feedback gain for linear quadratic optimal control that minimizes a worst-case CVaR of quadratic objective function for three cases; at the stationary distribution, at one-step cost and over an infinite time horizon. Such an objective function will minimize the expected value of the tail distribution of the quadratic cost.
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