The scenario approach is a general methodology for data-driven optimization that has attracted a great deal of attention in the past few years. It prescribes that one collects a record of previous cases (scenarios) from the same setup in which optimization is being conducted and makes a decision which attains optimality for the seen cases. Scenario optimization is by now very well understood for convex problems, where a theory exists that rigorously certifies the generalization properties of the solution, that is, the ability of the solution to perform well in connection to new situations. This theory supports the scenario methodology and justifies its use. This paper considers nonconvex problems. While other contributions in the non-convex setup already exist, we here take a major departure from previous approaches. We suggest that the generalization level is evaluated only after the solution is found and its complexity in terms of the length of a support sub-sample (a notion precisely introduced in this paper) is assessed. As a consequence, the generalization level is stochastic and adjusted case by case to the available scenarios. This fact is key to obtain tight results. The approach adopted in this paper applies not only to optimization, but also to generic decision problems where the solution is obtained according to a rule which is not necessarily the optimization of a cost function. Accordingly, in our presentation we adopt a general stance of which optimization is just seen as a particular case.

We study the problem of receding horizon control for stochastic discrete-time systems with bounded control inputs and incomplete state information. Given a suitable choice of causal control policies, we first present a slight extension of the Kalman filter to estimate the state optimally in mean-square sense. We then show how to augment the underlying optimization problem with a negative drift-like constraint, yielding a second-order cone program to be solved periodically online. We prove that the receding horizon implementation of the resulting control policies renders the state of the overall system mean-square bounded under mild assumptions. We also discuss how some quantities required by the finite-horizon optimization problem can be computed off-line, thus reducing the on-line computation

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In this paper, we first describe a matricial Newton-type algorithm designed to solve the multivariable spectrum approximation problem. We then prove its global convergence. Finally, we apply this approximation procedure to multivariate spectral estimation, and test its effectiveness through simulation. Simulation shows that, in the case of short observation records, this method may provide a valid alternative to standard multivariable identification techniques such as Matlab's PEM and Matlab's N4SID

ABSTRACT. In this article we construct control policies that ensure bounded variance of a noisy marginally stable linear system in closed-loop. It is assumed that the noise sequence is a mutually independent sequence of random vectors, enters the dynamics affinely, and has bounded fourth moment. The magnitude of the control is required to be of the order of the first moment of the noise, and the policies we obtain are simple and computable. § 1. INTRODUCTION Stabilization of stochastic linear systems with bounded control inputs has attracted considerable attention over the years. This is due to the fact that incorporating bounds on the control is of paramount importance in practical applications; suboptimal control strategies such as receding-horizon control Hokayem et al., 2009], and rollout algorithms [Bertsekas, 2000], among others, were designed to incorporate such constraints with relative ease, and have become widespread in applications. However, the following question remains open: when is a linear system with possibly unbounded additive stochastic noise globally stabilizable with bounded inputs? In this article we shall provide sufficient conditions that give a positive answer to this question with minimal hypotheses.Bounded input control has a rich and important history in the control literature [Lin et al., 1996;Stoorvogel et al., 2007;Sussmann et al., 1994;Yang et al., 1997Yang et al., , 1992. The deterministic version of the bounded input stabilization problem was solved completely in a series of articles [Sussmann et al., 1994;Yang et al., 1992] In the presence of affine stochastic noise the linear system ( ) becomes x t+1 = Ax t + Bu t + w t , where (w t ) t∈ 0 is a collection of independent (but not necessarily identically distributed) random vectors in d with possibly inter-dependent components at each time t. With an arbitrary noise it is clearly not possible to ensure mean-square boundedness; for instance, if the noise has a spherically symmetric Cauchy distribution on d , then given any initial condition x 0 ∈ d , the second moment of x 1 does not even exist. Similarly, if the second moment of the noise becomes unbounded with time, it is not possible to control the second moment of the process (x t ) t∈ 0 . It is necessary to assume, at least, that the noise has bounded variance.

Probabilistic Computation Tree Logic (PCTL) is a well-known modal logic which has become a standard for expressing temporal properties of finite-state Markov chains in the context of automated model checking. In this paper, we consider PCTL for noncountable-space Markov chains, and we show that there is a substantial affinity between certain of its operators and problems of Dynamic Programming. We prove some basic properties of the solutions to the latter. We also provide two examples and demonstrate how recovery strategies in practical applications, which are naturally stated as reach-avoid problems, can be viewed as particular cases of PCTL formulas

We start by summarizing the state of the art in stabilization of stochastic linear systems with bounded inputs and highlight remaining open problems. We then report two new results concerning mean-square boundedness of a linear system with additive stochastic noise. The first states that, given any nonzero bound on the controls, it is possible to construct a policy with bounded memory requirements that renders a marginally stable stabilizable system mean-square bounded in closed-loop. The second states that it is not possible to ensure mean-square boundedness in closed-loop with a bounded control policy for systems affected by unbounded noise and having at least one eigenvalue outside the unit circle

This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problemà la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback-Leibler pseudo-distance, which gives rise to a convex optimization problem. After developing the variational analysis, we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem, based on the iteration of a nonlinear map in a bounded subset of the dual space. Our main result is the proof of local convergence of the latter, established as a consequence of the Central Manifold Theorem. Supported by numerical evidence, we conjecture that, in the mentioned bounded set, the convergence is actually global.Moment problems have a long history and have been at the heart of many mathematical and engineering problems in the past century, see, e.g., [34], [1] and the references therein. Only with recent developments of the above-mentioned research program, however, the parametrization of solutions in the presence of additional constraints on the complexity have been satisfactorily addressed [12]. This result, that has been possible thanks to a suitable variational formulation, is of key interest in control engineering. In fact, the well-known relation between moment problems and Nevanlinna-Pick interpolation problems allows for solutions of H ∞ control problems that include a bound on the complexity of the controller, which is of paramount practical importance [3], [9]. Similar considerations apply to the covariance extension problem [10], [26]. Among the other applications, we also mention signal and image processing [22], [23], [24], [25], and Biomedical Engineering [30]. These applications are based on a spectral estimation procedure that hinges on optimal approximation of spectral densities with linear integral constraints that may be viewed as constraints on a finite number of moments of the spectrum. The linear integral constraints represent a knowledge on the steady-state state covariance of a bank of filters that is designed in order to estimate the unknown spectral density Φ, while the to-be-approximated spectral density represents a prior knowledge on Φ, see Section II for more details. As discussed in [7], [26], [32], this optimal approximation leads to a tunable spectral estimation algorithm that provides high resolution estimates in prescribed frequency bands even in presence of a short record of observed data. An important feature of the above mentioned optimal approximation method is that the primal optimization problem can be solved in closed form and, as long as the prior spectral density is rational, yields a rational solution with an a priori bound on the complexity (McMillan degree).The numerical challenge, in practical applications, lays with the dual problem. In fact, the dual variable is an Hermitian matrix and, as discussed in [26], the reparametrization in vector form may lead to a loss of convexity. Moreover, the dual functional and it...