Although continuous density estimation has received abundant attention in the Bayesian nonparametrics literature, there is limited theory on multivariate mixed scale density estimation. In this note, we consider a general framework to jointly model continuous, count and categorical variables under a nonparametric prior, which is induced through rounding latent variables having an unknown density with respect to Lebesgue measure. For the proposed class of priors, we provide sufficient conditions for large support, strong consistency and rates of posterior contraction. These conditions allow one to convert sufficient conditions obtained in the setting of multivariate continuous density estimation to the mixed scale case. To illustrate the procedure a rounded multivariate nonparametric mixture of Gaussians is introduced and applied to a crime and communities dataset.
In this paper, a new robust iterative learning control (ILC) algorithm has been proposed for linear systems in the presence of iteration-varying parametric uncertainties. The robust ILC design is formulated as a min-max problem using a quadratic performance criterion subject to constraints of the control input update. An upper bound of the maximization problem is derived, then, the solution of the min-max problem is achieved by solving a minimization problem. Applying Lagrangian duality to this minimization problem results in a dual problem which can be reformulated as a convex optimization problem over linear matrix inequalities (LMIs). Next, we present an LMIbased algorithm for the robust ILC design and prove the convergence of the control input and the error. Finally, the proposed algorithm is applied to a distillation column to demonstrate its effectiveness.
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