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Convex Relaxations of a Probabilistically Robust Control Design Problem


In this work, we address the problem of designing probabilistic robust controllers for discrete-time systems whose objective is to reach and remain in a given target set with high probability. More precisely, given probability distributions for the initial state, uncertain parameters and disturbances, we develop algorithms for designing a control law that  
i) maximizes the probability of reaching the target set in N steps  ii) makes the target set robustly positively invariant.  To solve this problem, a sequence of convex relaxations is provided, whose optimal value is shown to converge to solution of the original problem. In other words, we provide a sequence of semidefinite programs of increasing dimension and complexity which can arbitrarily approximate the solution of the probabilistic robust control design problem.



Probabilistically Robust Control Design






Convex Equivalent Problem:







Example 1






Example 2





Example 3: Probabilistic Control as Volume Optimization

More information on volume optimization problem:


Papers

  • Ashkan Jasour and Constantino Lagoa ”Convex Constrained Semialgebraic Volume Optimization: Application in Systems and Control”, Under submission, 2016.

Presentation

  • Convex Relaxations of a Probabilistically Robust Control Design Problem [pdf]
          52st IEEE Conference on Decision and Control, Florence, Italy, 2013