  This paper presents a convex optimization-based solution to the design of
state-feedback controllers for solving the linear quadratic regulator (LQR)
problem of uncertain discrete-time systems with multiplicative noise. To
synthesize a tractable solution, the recently developed system level synthesis
(SLS) framework is leveraged. It is shown that SLS shifts the controller
synthesis task from the design of a robust controller to the design of the
entire set-valued closed-loop system responses. To this end, the closed-loop
system response is entirely characterized by probabilistic set-valued maps from
the additive noise to control actions and states. A bi-level convex
optimization over the achievable set-valued closed-loop responses is then
developed to optimize the expected value of the LQR cost against the worst-case
closed-loop system response. The solution to this robust optimization problem
may be too conservative since it aims at enforcing the design constraints for
all possible system realizations. To deal with this issue, the presented
optimization problem is next reformulated as a chance-constrained program (CCP)
in which the guarantees are not intended in a deterministic sense of
satisfaction against all possible closed-loop system responses, but are instead
intended in a probabilistic sense of satisfaction against all but a small
fraction of the system responses. To approximately solve the CCP without the
requirement of knowing the probabilistic description of the uncertainty in the
system matrices, the so-called scenario optimization approach is employed,
which provides probabilistic guarantees based on a finite number of system
realizations and results in a convex optimization program with moderate
computational complexity. Finally, numerical simulations are presented to
illustrate the theoretical findings.
