Xavier Bombois

Model validation for control and controller validation in a prediction error identification framework-Part I: theory

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 parameterized transfer functions, which we call prediction error (PE) 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.

Brief Robustness analysis tools for an uncertainty set obtained by prediction error identification

This paper presents a robust stability and performance analysis for an uncertainty set delivered by classical prediction error identification. This nonstandard uncertainty set, which is a set of parametrized transfer functions with a parameter vector in an ellipsoid, contains the true system at a certain probability level. Our robust stability result is a necessary and sufficient condition for the stabilization, by a given controller, of all systems in such uncertainty set. The main new technical contribution of this paper is our robust performance result: we show that the worst case performance achieved over all systems in such an uncertainty region is the solution of a convex optimization problem involving linear matrix inequality constraints. Note that we only consider single input-single output systems.

Identification and the Information Matrix: How to Get Just Sufficiently Rich?

In prediction error identification, the information matrix plays a central role. Specifically, when the system is in the model set, the covariance matrix of the parameter estimates converges asymptotically, up to a scaling factor, to the inverse of the information matrix. The existence of a finite covariance matrix thus depends on the positive definiteness of the information matrix, and the rate of convergence of the parameter estimate depends on its ¿size¿. The information matrix is also the key tool in the solution of optimal experiment design procedures, which have become a focus of recent attention. Introducing a geometric framework, we provide a complete analysis, for arbitrary model structures, of the minimum degree of richness required to guarantee the nonsingularity of the information matrix. We then particularize these results to all commonly used model structures, both in open loop and in closed loop. In a closed-loop setup, our results provide an unexpected and precisely quantifiable trade-off between controller degree and required degree of external excitation.

Identification of dynamic models in complex networks with prediction error methods : basic methods for consistent module estimates

Abstract The problem of identifying dynamical models on the basis of measurement data is usually considered in a classical open-loop or closed-loop setting. In this paper, this problem is generalized to dynamical systems that operate in a complex interconnection structure and the objective is to consistently identify the dynamics of a particular module in the network. For a known interconnection structure it is shown that the classical prediction error methods for closed-loop identification can be generalized to provide consistent model estimates, under specified experimental circumstances. Two classes of methods considered in this paper are the direct method and the joint-IO method that rely on consistent noise models, and indirect methods that rely on external excitation signals like two-stage and IV methods. Graph theoretical tools are presented to verify the topological conditions under which the several methods lead to consistent module estimates.

Cheapest open-loop identification for control

This paper presents a new method of identification experiment design for control. Our objective is to design the open-loop identification experiment with minimal excitation such that the controller designed with the identified model stabilizes and achieves a prescribed level of H/sub /spl infin// performance with the unknown true system G/sub 0/.

A measure of robust stability for an identified set of parametrized transfer functions

Defines a measure of robustness for a set of parameterized transfer functions as delivered by classical prediction error identification and that contains the true system at a prescribed probability level. This measure of robustness is the worst case Vinnicombe distance between the model and the plants in the uncertainty region. We show how it can be computed exactly using LMI-based optimization. In addition, we show that this measure is directly connected to the size of the set of controllers that are guaranteed to stabilize all plants in the uncertainty region, i.e., the smaller the worst case Vinnicombe distance for an uncertainty region, the larger the set of model-based controllers that are guaranteed to stabilize all systems in this uncertainty region.

Controller validation based on an identified model

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.

Identification of Dynamic Models in Complex Networks With Prediction Error Methods: Predictor Input Selection

This paper addresses the problem of obtaining an estimate of a particular module of interest that is embedded in a dynamic network with known interconnection structure. In this paper it is shown that there is considerable freedom as to which variables can be included as inputs to the predictor, while still obtaining consistent estimates of the particular module of interest. This freedom is encoded into sufficient conditions on the set of predictor inputs that allow for consistent identification of the module. The conditions can be used to design a sensor placement scheme, or to determine whether it is possible to obtain consistent estimates while refraining from measuring particular variables in the network. As identification methods the Direct and Two Stage Prediction-Error methods are considered. Algorithms are presented for checking the conditions using tools from graph theory.

Input design: from open-loop to control-oriented design

In this paper we briefly review the evolution of the main tools and results for optimal experiment design for system identification. The initial work dates back to the seventies and focused on the accuracy of the parameters of the input-output transfer function estimate. In the eighties, new formulas for the variance of transfer function estimates based on high-order model approximations led to the first goal-oriented experiment design results. The recent trend is to address control-oriented optimal design questions using the more accurate parameter covariance formulas for finite order models.

Data-driven model improvement for model-based control

We present a framework for the gradual improvement of model-based controllers. The total time of the learning procedure is divided into a number of learning intervals. After a learning interval, the model is refined based on the measured data. This model is used to synthesize the controller that will be applied during the next learning interval. Excitation signals can be injected into the control loop during each of the learning intervals. On the one hand, the introduction of an excitation signal worsens the control performance during the current learning interval since it acts as a disturbance. On the other hand, the informative data generated owing to the excitation signal are used to refine the model using a closed-loop system identification technique. Therefore, the control performance for the next learning interval is expected to improve. In principle, our objective is to maximize the overall control performance taking the effect of the excitation signals explicitly into account. However, this is in general an intractable optimization problem. For this reason, a convex approximation of the original problem is derived using standard relaxations techniques for Experiment Design. The approximated problem can be solved efficiently using common optimization routines. The applicability of the method is demonstrated in a simulation study.