Marco C. Campi

Brief Virtual reference feedback tuning: a direct method for the design of feedback controllers

This paper considers the problem of designing a controller for an unknown plant based on input/output measurements. The new design method we propose is direct (no model identification of the plant is needed) and can be applied using a single set of data generated by the plant, with no need for specific experiments nor iterations. It is shown that the method searches for the global optimum of the design criterion and that, in the case of restricted complexity controller design, the achieved controller is a good approximation of the restricted complexity global optimal controller. A simulation example shows the effectiveness of the method.

Uncertain convex programs: randomized solutions and confidence levels

Abstract.Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or ‘instance’ parameter. Two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty: robust optimization and chance-constrained optimization. Robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances. In chance-constrained optimization a probability distribution is instead assumed on the uncertain parameters, and the constraints are enforced up to a pre-specified level of probability. Unfortunately however, both approaches lead to computationally intractable problem formulations.In this paper, we consider an alternative ‘randomized’ or ‘scenario’ approach for dealing with uncertainty in optimization, based on constraint sampling. In particular, we study the constrained optimization problem resulting by taking into account only a finite set of N constraints, chosen at random among the possible constraint instances of the uncertain problem. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient and explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.

The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs

Many optimization problems are naturally delivered in an uncertain framework, and one would like to exercise prudence against the uncertainty elements present in the problem. In previous contributions, it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization. In this paper, we establish new feasibility results for randomized algorithms. Specifically, the exact feasibility for the class of the so-called fully-supported problems is obtained. It turns out that all fully-supported problems share the same feasibility properties, revealing a deep kinship among problems of this class. It is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully-supported problems. The feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully-supported problems.

A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality

In this paper, we study the link between a Chance-Constrained optimization Problem (CCP) and its sample counterpart (SP). SP has a finite number, say N, of sampled constraints. Further, some of these sampled constraints, say k, are discarded, and the final solution is indicated by

The scenario approach for systems and control design

x^{\ast}_{N,k}

Direct nonlinear control design: the virtual reference feedback tuning (VRFT) approach

. Extending previous results on the feasibility of sample convex optimization programs, we establish the feasibility of

An application of the virtual reference feedback tuning method to a benchmark problem

x^{\ast}_{N,k}

Unbiased estimation of a sinusoid in colored noise via adapted notch filters

for the initial CCP problem.Constraints removal allows one to improve the cost function at the price of a decreased feasibility. The cost improvement can be inspected directly from the optimization result, while the theory here developed permits to keep control on the other side of the coin, the feasibility of the obtained solution. In this way, trading feasibility for performance is put on solid mathematical grounds in this paper.The feasibility result here obtained applies to a vast class of chance-constrained optimization problems, and has the distinctive feature that it holds true irrespective of the algorithm used to discard k constraints in the SP problem. For constraints discarding, one can thus, e.g., resort to one of the many methods introduced in the literature to solve chance-constrained problems with discrete distribution, or even use a greedy algorithm, which is computationally very low-demanding, and the feasibility result remains intact.We further prove that, if constraints in the SP problem are optimally removed—i.e., one deletes those constraints leading to the largest possible cost improvement—, then a precise optimality link to the original chance-constrained problem CCP in addition holds.

Interval predictor models: Identification and reliability

The ‘scenario approach’ is an innovative technology that has been introduced to solve convex optimization problems with an infinite number of constraints, a class of problems which often occurs when dealing with uncertainty. This technology relies on random sampling of constraints, and provides a powerful means for solving a variety of design problems in systems and control. The objective of this paper is to illustrate the scenario approach at a tutorial level, focusing mainly on algorithmic aspects. Its versatility and virtues will be pointed out through a number of examples in model reduction, robust and optimal control.

Virtual reference feedback tuning (VRFT): a new direct approach to the design of feedback controllers

This paper introduces the virtual reference feedback tuning (VRFT) approach for controller tuning in a nonlinear setup. VRFT is a data-based method that permits to directly select the controller based on data, with no need for a model of the plant. It is based on a global model reference optimization procedure and, therefore, does not require to access the plant for experiments many times so as to estimate the control cost gradient. For this reason, it represents a very appealing controller design methodology for many control applications.