Data validation of uncertain dynamic systems

Automatica, 2003

We propose a model validation procedure that consists of a prediction error identification experiment with a full order model. It delivers a parametric uncertainty ellipsoid and a corresponding set of parametrized transfer functions, which we call PE (for Prediction Error) uncertainty set. Such uncertainty set differs from the classical uncertainty descriptions used in robust control analysis and design. We develop a robust control analysis theory for such uncertainty sets, which covers two distinct aspects. (1) Controller validation. We present necessary and sufficient conditions for a specific controller to stabilize -or to achieve a given level of performance with -all systems in such PE uncertainty set. (2) Model validation for robust control. We present a measure for the size of such PE uncertainty set that is directly connected to the size of a set controllers that stabilize all systems in the model uncertainty set. This allows us to establish that one uncertainty set is better tuned for robust control design than another, leading to control-oriented validation objectives.

1996

Abstract We consider time-domain model validation for sampled-data systems with H∞-compatible uncertainty models. These uncertainty models consist of a nominal continuous-time plant model together with given bounds for system uncertainty and signal disturbances. The validation problem is to determine whether or not a given discrete sampled input-output data record is consistent with a postulated uncertainty model.

IEEE Transactions on Automatic Control, 1994

Abstruct-In this paper we offer a novel approach to controloriented model validation problems. The problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input-output data. Our papproach directly uses time-domain input-output data to validate uncertainty models. The algorithms we develop are computationally tractable and reduce to (generally non-differentiable) convex feasibility problems or to linear programming problems. In special cases, we give analytical solutions to these problems.

IFAC Proceedings Volumes (IFAC-PapersOnline), 2009

This paper proposes a data-driven test for closed-loop stability. The test is based on a non-conservative stability condition that can be verified without having to actually implement the controller. It uses a set of measurements from the plant but does not rely on a plant model. For infinite data length, a validated controller is guaranteed to stabilize the plant. In practice, however, only a finite number of noisy data can be used, and thus only an estimate of the stability condition can be obtained. A reliable stability test needs to take this estimation uncertainty into account, which introduces conservatism. In the proposed test, two variables are available to control the trade-off between reliability and conservatism in an intuitive way. A simulation example shows the effectiveness of the stability test.

Computer Methods in Applied Mechanics and Engineering, 2008

We present a methodology to perform the identification and validation of complex uncertain dynamical systems using experimental data, for which uncertainties are taken into account by using the nonparametric probabilistic approach. Such a probabilistic model of uncertainties allows both model uncertainties and parameter uncertainties to be addressed by using only a small number of unknown identification parameters. Consequently, the optimization problem which has to be solved in order to identify the unknown identification parameters from experiments is feasible. Two formulations are proposed. The first one is the mean-square method for which a usual differentiable objective function and an unusual non-differentiable objective function are proposed. The second one is the maximum likelihood method coupling with a statistical reduction which leads us to a considerable improvement of the method. Three applications with experimental validations are presented in the area of structural vibrations and vibroacoustics.

Proceedings of the 19th IFAC World Congress, 2014

Probabilistic and set-based methods are two approaches for model (in)validation, parameter and state estimation. Both classes of methods use different types of data, i.e. deterministic or probabilistic data, which allow different statements and applications. Ideally, however, all available data should be used in estimation and model invalidation methods. This paper presents an estimation and model (in)validation framework combining set-based and probabilistically uncertain data for polynomial continuous-time systems. In particular, uncertain data on the moments and the support is used without the need to make explicit assumptions on the type of probability densities. The paper derives pointwise-in-time outer approximations of the moments of the probability densities associated with the states and parameters of the system. These approximations can be interpreted as guaranteed confidence intervals for the moment estimates. Furthermore, guaranteed bounds on the probability masses on subsets are derived and allow an estimation of the unknown probability densities. To calculate the estimates, the dynamics of the probability densities of the state trajectories are found by occupation measures of the nonlinear dynamics. This allows the construction of an infinite-dimensional linear program which incorporates the set-and moment-based data. This linear program is relaxed by a hierarchy of LMI problems providing, as shown elsewhere, an almost uniformly convergent sequence of outer approximations of the estimated sets. The approach is demonstrated with numerical examples.

… 1999. Proceedings of the 38th IEEE …, 1999

Static linear models characterized by bounded uncertainties on both the equation error and the parameters are studied. The additive equation error is assumed to belong to an interval while the parameters fluctuate inside a time-invariant bounded domain. An algorithm is proposed for evaluating different bounded domains. The algorithm can be extended to cope with the determination of the central value of the domain containing the parameters. Contrary to most of traditional estimators, the resulting estimator takes into account the distribution of the uncertainties of a model.

Starting with our work on the characterization of the imprecision error in programs using floating-point-numbers, by abstract interpretation, this paper shows that there are numerous perspectives, if one wants to fully qualify the numerical quality of control systems, as found in the aeronautical and automotive industry, for instance. Some very common functions (e.g. integrators) are hard to statically analyse, because their numerical correctness depend on a finegrained specification of the classes of input signals they handle. This gets even more complex in the case of e.g. PID controllers, which interact in closed loop with an external environment, since their input signals are in part the consequence of their own computation, similarly for the imprecision errors. We show examples of non-trivial bad and good numerical behaviours, discuss the results of our methods, and present our current research directions, that should hopefully help characterize the imprecision error of such control systems.

AIAA Journal, 2004

Model validation has become a primary means to evaluate accuracy and reliability of computational simulations in engineering design. Due to uncertainties involved in modeling, manufacturing processes, and measurement systems, the assessment of the validity of a modeling approach must be conducted based on stochastic measurements to provide designers with the confidence of using a model. In this paper, a generic model validation methodology via uncertainty propagation and data transformations is presented. The approach reduces the number of physical tests at each design setting to one by shifting the evaluation effort to uncertainty propagation of the computational model. Response surface methodology is used to create metamodels as less costly approximations of simulation models for the uncertainty propagation. Methods for validating models with both normal and nonnormal response distributions are proposed. The methodology is illustrated with the examination of the validity of two finite element analysis models for predicting springback angles in a sample flanging process.

Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), 1999

This paper focuses on the validation of a controller that has been designed from an unbiased model of the true system, identified either in open-loop or in closed-loop using a prediction error framework. A controller is said to be validated if it stabilizes all models in a parametric uncertainty set containing the parameters of the true system with some prescribed probability. This uncertainty set is deduced from the covariance matrix of the parameters of the identified model. Our contribution is to embed this set in the smallest possible overbounding coprime factor uncertainty set. This then allows us to use the results of mainstream robust control theory such as the Vinnicombe gap between plants and its related stability theorems.