Mario Milanese

Optimal estimation theory for dynamic systems with set membership uncertainty: an overview

In many problems such as linear and nonlinear regressions, parameter and state estimation of dynamic systems, state-space and time series prediction, interpolation, smoothing and functions approximation, one has to evaluate some unknown variable using available data. The data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables. Typically, the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory. An interesting alternative approach, referred to as set membership or unknown but bounded (UBB) error description, has been investigated since the late 1960s. In this approach, uncertainty is described by an additive noise which is known only to have given integral (typically l2 of l1) or componentwise (l∞) bounds. In this paper we review the main results of this theory, with special attention to the most recent advances obtained in the case of componentwise bounds.

Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: Linear families of models and estimators

The problem of parameter estimation and of the evaluation of related uncertainty intervals is considered in the case that a probabilistic description of noise and errors is not available (of suitable), but only a bound on them is known. In the present paper only linear parametrizations and estimators are considered. Very simple and computationally feasible algorithms are derived for evaluating two different types of uncertainty intervals (Estimates Uncertainty Intervals, Parameter Uncertainty Intervals). The relationships between the EUIs and the PUIs are established, and the solution to the problem of the minimum uncertainty intervals estimator is given: the latter can be obtained by means of a simple linear programming algorithm.

Optimal algorithms theory for robust estimation and prediction

This paper deals with the theory of optimal algorithms for problems which cannot be solved exactly. The theory developed allows for the derivation of new and interesting results in parameter estimation and in time series prediction in situations where no reliable statistical hypothesis can be made on the functions and modeling errors involved, but only a bound on them is known, in particular, the derivation of computationally simple optimal algorithms for these two problems is investigated. The practical effectiveness of the algorithms obtained is illustrated by several numerical examples.

Description and simulation of a physiological pharmacokinetic model for the metabolism and enterohepatic circulation of bile acids in man. Cholic acid in healthy man.

A multicompartmental pharmacokinetic model based on physiological principles, experimental data, and the standard mathematical principles of compartmental analysis has been constructed that fully describes the metabolism and enterohepatic cycling in man of cholic acid, a major bile acid. The model features compartments and linear transfer coefficients. The compartments are aggregated into nine spaces based on physiological considerations (liver, gallbladder, bile ducts, jejunum, ileum, colon, portal blood sinusoidal blood, and general circulation). The transfer coefficients are also categorized according to function: flow, i.e., emptying of gallbladder or intestinal spaces, and circulation of the blood; biotransformation, i.e., conjugation, deconjugation, or dehydroxylation; and transport, i.e., active or passive transport. The model is made time dependent by introducing meals, which trigger discrete increases in gallbladder emptying and intestinal flow. Each space contains three compartments. For cholic acid, these are unconjugated cholic acid, cholylglycine, and cholyltaurine. The model was then used with all existing experimental data to simulate cholic acid metabolism in healthy man over a 24-h period. Satisfactory agreement was obtained between simulated and experimental results for serum bile acid levels, hepatic bile acid secretion, and bile acid secretion into the intestine. The model was also used to classify 16 clinical instances in which the enterohepatic circulation of bile acids is altered by drugs or disease. The model can be extended to describe completely the metabolism and enterohepatic circulation of any bile acids in man in health and digestive disease. The model should also be broadly applicable to the description of the pharmacokinetics of all other drugs whose metabolism is similar to that of bile acids, i.e., drugs for which there are tissue and bacterial biotransformations, enterohepatic cycling, and appreciable first-pass clearance.

Estimation theory for nonlinear models and set membership uncertainty

This chapter studies the problem of estimating a given function of a vector of unknowns, called the problem element, by using measurements depending non-linearly on the problem element and affected by unknown but bounded noise. Assuming that both the solution sought and the measurements depend polynomially on the unknown problem element, a method is given to compute the axis-aligned box of minimal volume containing the feasible solution set, i.e., the set of all unknowns consistent with the actual measurements and the given bound on the noise. The center of this box is a point estimate of the solution, which enjoys useful optimality properties. The sides of the box represent the intervals of possible variation of the estimates. Important problems, like parameter estimation of exponential models, time series prediction with ARMA models and parameter estimates of discrete time state space models, can be formalized and solved by using the developed theory.

Optimality of central and projection algorithms for bounded uncertainty

This paper investigates optimality properties of central and projection algorithms for linear problems in the field of system identification in a context in which uncertainty is described in a deterministic rather than statistical way. Particular attention is devoted to least-squares algorithms when the measurement noise is assumed to be unknown but bounded in a Hilbert norm. A major contribution of this paper consists in proving that least-squares algorithms enjoy strong optimality properties. On the contrary, it is pointed out that these properties do not hold for other frequently used projection algorithms, such as least-absolute-values or minimax algorithms, corresponding to a description of the measurement error in h or l∞ norm, respectively.

Combined Automatic Lane-Keeping and Driver's Steering Through a 2-DOF Control Strategy

In this paper, we address the problem of combining automatic lane-keeping and drivers steering for either obstacle avoidance or lane-change maneuvers for passing purposes or any other desired maneuvers, through a closed-loop control strategy. The automatic lane-keeping control loop is never opened, and no on/off switching strategy is used. During the drivers maneuver, the vehicle lateral dynamics are controlled by the driver himself through the vehicle steering system. When there is no drivers steering action, the vehicle center of gravity tracks the center of the traveling lane thanks to the automatic lane-keeping system. At the beginning (end) of the maneuver, the lane-keeping task is released (resumed) safely and smoothly. The performance of the proposed closed-loop structure is shown both by means of simulations and through experimental results obtained along Italian highways.

Computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties

Consideration is given to the problem of robust stability analysis of linear dynamic systems with uncertain physical parameters entering as polynomials in the state equation matrices. A method is proposed giving necessary and sufficient conditions for computing the uncertain system stability margin in parameter space, which provides a measure of maximal parameter perturbations preserving stability of the perturbed system around a known, stable, nominal system. A globally convergent optimization algorithm that enables solutions to the problem to be obtained is presented. Using the polynomial structure of the problem, the algorithm generates a convergent sequence of interval estimates of the global extremum. These intervals provide a measure of the accuracy of the approximating solution achieved at each step of the iterative procedure. Some numerical examples are reported, showing attractive features of the algorithm from the point of view of computational burden and convergence behavior. >

Robust analysis and design of control systems using interval arithmetic

Abstract Several robustness problems such as stability and performance robustness analysis of feedback systems and robust design of control systems in the presence of mixed nonlinear parametric and nonparametric perturbations can be solved by means of algorithms based on interval-arithmetic computation. Some of the main algorithms available in the literature are presented, and their efficiency is tested on some examples of robustness analysis and design of control systems.

Airborne Wind Energy: An overview

In the last decade, several research groups and companies around the world have been developing a new class of wind generators, aimed at harnessing the energy of winds blowing at high elevation above the ground. This kind of technology is usually referred to as Airborne Wind Energy (AWE) or High-Altitude Wind Energy. All of the proposed solutions exploit the high-speed flight of tethered wings, or aircrafts, and their operation heavily relies on automatic control. This paper provides a tutorial on the fundamental concepts of AWE and on the different technologies that are being investigated, with particular emphasis on control-related aspects, highlighting the accomplished results and the issues that still need to be solved.