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We develop a model-free learning algorithm for the infinite-horizon linearquadratic regulator (LQR) problem. Specifically, (risk) constraints andstructured feedback are considered, in order to reduce the state deviationwhile allowing for a sparse communication graph in practice. By reformulatingthe dual problem as a nonconvex-concave minimax problem, we adopt the gradientdescent max-oracle (GDmax), and for modelfree setting, the stochastic (S)GDmaxusing zero-order policy gradient. By bounding the Lipschitz and smoothnessconstants of the LQR cost using specifically defined sublevel sets, we candesign the stepsize and related parameters to establish convergence to astationary point (at a high probability). Numerical tests in a networkedmicrogrid control problem have validated the convergence of our proposed SGDmaxalgorithm while demonstrating the effectiveness of risk constraints. The SGDmaxalgorithm has attained a satisfactory optimality gap compared to the classicalLQR control, especially for the full feedback case.

Model-free Learning for Risk-constrained Linear Quadratic Regulator with Structured Feedback in Networked Systems