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  4. Semidefinite programs (SDP) are a staple of today’s systems theory, with applications ranging from robust control to systems identification. However, current state-of-the art solution methods have poor scaling properties, and thus are limited to relatively moderate size problems. Recently, several approximations have been proposed where the original SDP is relaxed to a sequence of lower complexity problems (such as linear programs (LPs) or second order cone programs (SOCPs)). While successful in many cases, there is no guarantee that these relaxations converge to the global optimum of the original program. Indeed, examples exists where these relaxations "get stuck" at suboptimal solutions. To circumvent this difficulty in this paper we propose an algorithm to solve SDPs based on solving a sequence of LPs or SOCPs, guaranteed to converge in a finite number of steps to an ε-suboptimal solution of the original problem. We further provide a bound on the number of steps required, as a function of ε and the problem data.
    Free, publicly-accessible full text available December 14, 2022
  5. Peak estimation bounds extreme values of a function of state along trajectories of a dynamical system. This paper focuses on extending peak estimation to continuous and discrete settings with time-independent and time-dependent uncertainty. Techniques from optimal control are used to incorporate uncertainty into an existing occupation measure-based peak estimation framework, which includes special consideration for handling switching-type (polytopic) uncertainties. The resulting infinite-dimensional linear programs can be solved approximately with Linear Matrix Inequalities arising from the moment-SOS hierarchy.
    Free, publicly-accessible full text available December 14, 2022
  6. Recently, there has been renewed interest in data-driven control, that is, the design of controllers directly from observed data. In the case of linear time-invariant (LTI) systems, several approaches have been proposed that lead to tractable optimization problems. On the other hand, the case of nonlinear dynamics is considerably less developed, with existing approaches limited to at most rational dynamics and requiring the solution to a computationally expensive Sum of Squares (SoS) optimization. Since SoS problems typically scale combinatorially with the size of the problem, these approaches are limited to relatively low order systems. In this paper, we propose an alternative, based on the use of state-dependent representations. This idea allows for synthesizing data-driven controllers by solving at each time step an on-line optimization problem whose complexity is comparable to the LTI case. Further, the proposed approach is not limited to rational dynamics. The main result of the paper shows that the feasibility of this on-line optimization problem guarantees that the proposed controller renders the origin a globally asymptotically stable equilibrium point of the closed-loop system. These results are illustrated with some simple examples. The paper concludes by briefly discussing the prospects for adding performance criteria.
    Free, publicly-accessible full text available December 14, 2022
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