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Performance bounds and suboptimal policies for linear stochastic control

Yang Wang and Stephen Boyd

In this talk we consider a causal state feedback stochastic control problem with linear dynamics, arbitrary noise distribution, and arbitrary input constraint set.  We present several widely-used suboptimal control policies and compare their performance to a lower bound on optimal performance, which we compute by solving a semi-definite program. Numerical experiments show that the lower bound obtained by our method is often close to the performance achieved by these policies, which shows that both are nearly optimal. As a by-product, our performance bound yields approximate value functions that work very well as control-Lyapunov functions (or terminal costs) for suboptimal controllers. These controllers can also be implemented online at rates exceeding tens of kilohertz, which makes them entirely practical for a wide range of problems.