Sippe G. Douma

Relations between uncertainty structures in identification for robust control

Various techniques of system identification exist that provide a nominal model and an uncertainty bound. An important question is what the implications are for the particular choice of the structure in which the uncertainty is described when dealing with robust stability/performance analysis of a given controller and when dealing with robust synthesis. It is shown that an amplitude-bounded (circular) uncertainty set can equivalently be described in terms of an additive, Youla parameter and @n-gap uncertainty. As a result, the choice of structure does not matter provided that the identification methods deliver optimal uncertainty sets rather than an uncertainty bound around a prefixed nominal model. Frequency-dependent closed-loop performance functions based on the uncertainty sets are again bounded by circles in the frequency domain, allowing for analytical expressions for worst-case performance and for the evaluation of the consequences of uncertainty for robust design. The results can be used to tune optimal experimental conditions in view of robust control design and in the further development of experiment-based robust control design methods.

Brief Controller tuning freedom under plant identification uncertainty: double Youla beats gap in robust stability

In iterative schemes of identification and control one of the particular and important choices to make is the choice for a model uncertainty structure, capturing the uncertainty concerning the estimated plant model. Structures that are used in the recent literature encompass e.g. gap metric uncertainty, coprime factor uncertainty, and the Vinnicombe gap metric uncertainty. In this paper, we study the effect of these choices by comparing the sets of controllers that guarantee robust stability for the different model uncertainty bounds. In general these controller sets intersect. However in particular cases the controller sets are embedded, leading to uncertainty structures that are favourable over others. In particular, when restricting the controller set to be constructed as metric-bounded perturbations around the present controller, the so-called double Youla parametrization provides a set of robustly stabilizing controllers that is larger than corresponding sets that are achieved by using any of the other uncertainty structures. This is particularly of interest in controller tuning problems.

Relation between uncertainty structures in identification for robust control

Abstract Various techniques of system identification exist providing for a nominal model and uncertainty bound. An important question is what the implications are for the particular choice of the structure in which the uncertainty is described when dealing with robust stability/performance analysis of a given controller and when dealing with robust synthesis. An amplitude-bounded (circular) uncertainty set can equivalently be described in terms of an additive, Youla parameter and v-gap uncertainty. Closed-loop performance functions based on these sets are again bounded by circles in the frequency domain, allowing for exact worst-case performance calculation and for the evaluation of the consequences of uncertainty for robust design.

PROBABILISTIC MODEL UNCERTAINTY BOUNDING: AN APPROACH WITH FINITE-TIME PERSPECTIVES

Abstract In prediction error identification model uncertainty bounds are generally derived from the statistical properties of the parameter estimator, i.e. asymptotic normal distribution of the estimator, and availability of the covariance information. When the primal interest of the identification is in a-posteriori quantifying the uncertainty in an estimated parameter, alternative parameter confidence bounds can be constructed. Probabilistic parameter confidence bounds are studied for ARX models which are generated by computationally more simple expressions, and which have the potential of being less dependent on asymptotic approximations and assumptions. It is illustrated that the alternative bounds can be powerful for quantifying parameter confidence regions for finite-time situations.

Validity of the standard cross-correlation test for model structure validation

Abstract The standard prediction error framework provides many theoretical results under the assumption that the true system is in the model class. An important example is the expression for the parameter covariance matrix which is used to derive model uncertainty regions. An essential step in a system identification procedure is the (in)validation of this assumption that the model structure is rich enough to contain the true system. The standard test for this purpose is the sample cross-correlation test between the output residuals and the input. It turns out that this standard test itself is valid only under exactly those assumptions it is meant to verify. As a result considerable undermodelling errors can remain undetected. Besides suggesting caution to users of the standard test, methods are presented to adapt the test adequately.

CONTROLLER TUNING FREEDOM UNDER PLANT IDENTIFICATION UNCERTAINTY: DOUBLE YOULA BEATS GAP IN ROBUST STABILITY

Abstract In iterative schemes of identification and control one of the particular and important choices to make is the choice for a model uncertainty structure, capturing the uncertainty concerning the estimated plant model. This is typically done in some norm-bounded form, in order to guarantee robust stability and/or robust performance when redesigning the controller. Structures that are used in the recent literature encompass e.g. gap metric uncertainty, coprime factor uncertainty, and the Vinnicombe gap metric uncertainty. In this paper we study the effect of these choices when our aim is to maximize the (re)tuning freedom for a present controller (in terms of a norm-bounded perturbation) under conditions of robust stability. Particular attention will be given to the representation of plant uncertainty and controller tuning freedom in terms of Youla parameters. In the problem formulation considered here the so-called double Youla parametrization provides a norm-bounded set of robustly stabilizing controllers that is larger than corresponding sets that are achieved by using any of the other uncertainty structures. Copyright & 2002 IFAC

An Alternative Paradigm for Probabilistic Uncertainty Bounding in Prediction Error Identification

In prediction error identification model uncertainty bounds are generally derived from the statistical properties of the parameter estimator. These statistical properties reflect the variability in the estimated model under repetition of experiments with different realizations of the measured signals. However when the primal interest of the identification is in quantifying the uncertainty in an estimated parameter on the basis of one single experiment, this is not necessarily the best and only approach. In the alternative paradigm that is presented here, not the covariance of the estimator will be used for bounding the model uncertainty, but an a posteriori bound on the error in the estimated parameter will be constructed that is structurally dependent on the particular data sequence. This will allow simpler computations for probabilistic model uncertainty bounds also applicable to the situation of approximate modelling (S ∉ M) and to model structures that are nonlinear in the parameters, such as Output Error (OE) models.

On the choice of uncertainty structure in identification for robust control

Various techniques of system identification exist providing for a nominal model and uncertainty bound. An important question is what the implications are for the particular choice of the structure in which the uncertainty is described when dealing with robust stability/performance analysis of a given controller and when dealing with robust synthesis. An amplitude-bounded (circular) uncertainty set can equivalently be described in terms of an additive, Youla parameter and /spl nu/-gap uncertainty. Closed-loop performance functions based on these sets are again bounded by circles in the frequency domain, allowing for exact worst-case performance calculation and for the evaluation of the consequences of uncertainty for robust design.

Controller tuning freedom under plant identification uncertainty: double Youla beats gap in robust stability

In iterative schemes of identification and control one of the particular and important choices to make is the choice for a model uncertainty structure, capturing the uncertainty concerning the estimated plant model. This is typically done in some norm-bounded form, in order to guarantee robust stability and/or robust performance when re-designing the controller. Structures that are used in the recent literature encompass include the gap metric uncertainty, coprime factor uncertainty, and Vinnicombe gap metric uncertainty. In this paper we study the effect of these choices when our aim is to maximize the (re)tuning freedom for a present controller under conditions of robust stability. Particular attention is given to the representation of plant uncertainty and controller tuning freedom in terms of Youla parameters. This so-called double Youla parametrization provides a norm-bounded set of robustly stabilizing controllers that is larger than corresponding sets that are achieved by using any of the other uncertainty structures.

Brief paper: Validity of the standard cross-correlation test for model structure validation

In the standard prediction error framework of system identification, statistical properties of estimated models are typically derived under the assumption that the true system is in the model class. The standard model structure validation test for plant models is the sample cross-correlation test between the residuals of the model and the input. It turns out that the standard test itself is valid only under exactly those assumptions it is meant to verify, i.e. the system is in the model class. It is shown that for reliable results of the validation test a vector-valued test is required and that accurate noise modelling is indispensable for reliable model structure validation. This shows the limitation of separate validation of plant and noise model structures. Improvements of the test are presented, and it is motivated by the fact that reserving data only to be used for model validation is not efficient.