  We propose a new risk-constrained reformulation of the standard Linear
Quadratic Regulator (LQR) problem. Our framework is motivated by the fact that
the classical (risk-neutral) LQR controller, although optimal in expectation,
might be ineffective under relatively infrequent, yet statistically significant
(risky) events. To effectively trade between average and extreme event
performance, we introduce a new risk constraint, which explicitly restricts the
total expected predictive variance of the state penalty by a user-prescribed
level. We show that, under rather minimal conditions on the process noise
(i.e., finite fourth-order moments), the optimal risk-aware controller can be
evaluated explicitly and in closed form. In fact, it is affine relative to the
state, and is always internally stable regardless of parameter tuning. Our new
risk-aware controller: i) pushes the state away from directions where the noise
exhibits heavy tails, by exploiting the third-order moment (skewness) of the
noise; ii) inflates the state penalty in riskier directions, where both the
noise covariance and the state penalty are simultaneously large. The properties
of the proposed risk-aware LQR framework are also illustrated via indicative
numerical examples.
