System Identification and Robust Control: A Synergistic Approach

The main purpose of this work is to develop a coherent system identification based robust control design methodology by combining recent results from system identification and robust control. In order to accomplish this task new theoretical results will be given in both fields. Firstly, however, an introduction to modern robust control design analysis and synthesis will be given. It will be shown how the classical frequency domain techniques can be extended to multivariable systems using the singular value decomposition. An introduction to norms and spaces frequently used in modern control theory will be given. However, the main emphasis in this thesis will not be on mathematics. Proofs are given only when there are interesting in their own right and we will try to avoid messy mathematical derivations. Rather we will concentrate on interpretations and practical design issues. A review of the classical H infinity theory will be given. Most of the stability results from modern control theory can be traced back to multivariable generalization of the famous Nyquist stability criterion is used to establish some of the classical singular value robust stability results. Furthermore it will be shown how a performance specification may be cast into the same framework. The main limitation in the standard H infinity theory is that it can only handle unstructured complex full block perturbations to the nominal plant. However, often much more detailed perturbation models are available, e.g. from physical modelling. Such structured perturbation models cannot be handled well in the H infinity framework. Fortunately theory exists which can do this. The structured singular value m is an extension of the singular value, which explicitly takes into account the structure of the perturbations. In this thesis we will present a thorough introduction to the m framework. A central results is that if performance is measured in terms of the inifity-norm and model uncertainty is bounded in the same manner, then, using m it is possible to pose one necessary and sufficient condition for block-structured norm-bounded perturbations which enter the nominal model in a linear fractional manner. This is, however, a very general perturbation set which includes a large variety of uncertainty such as unstructured and structured dynamic uncertainty (complex perturbations) and parameter variations (real perturbations). The uncertainty structures permitted by m is definitely much more flexible than those used in H inifity. Unfortunately m synthesis is a very difficult mathematical problem which is only well developed for purely complex perturbation sets. In order to develop our main result we will unfortunately need to synthesize m controllers for mixed real and complex perturbation sets. A novel method, denoted m - K iteration, has been develop to solve the mixed m problem. A general feature of all robust control design methods is the need for specifying not only a nominal model but also some kind of quantification of the uncertainty is, however, a non-trivial problem which to some extent has been neglected by the theoreticians of robust control. An uncertainty specification has simply been assumed given. One way of obtaining a perturbation model is by physical modelling. Application if the fundamental laws of thermodynamics, mechanics, physics etc. Will generally yield a set of coupled non-linear partial differential equations. These equations can then be linearized (in time and position) around a suitable working point and Laplace transformed for linear control design. The linearized differential equations will typically involve physical quantities like masses, inertias, etc. Which are only known with a certain degree of accuracy. This will give rise to real scalar perturbation to the nominal model. Furthermore working point deviations may also be addressed with real perturbations. However, accurate physical modelling may be a complicated and time consuming task even for relatively simple systems. An appealing alternative to physical modelling for assessment of model uncertainty is system identification where input/output measurements are used to estimate a, typically, linear model of the true system. From the covariance of the parameter estimate frequency domain uncertainty estimates may be obtained. In classical (i.e. Ljungian) system identification, model quality has been assessed under the structure of the model is assumed to be correct. This is, however, often an inadequate assumption in connection with control design. Recently, system identification techniques for estimating model uncertainty have gained renewed interest in the automatic control community. N this thesis, a quick survey of these results will be given together with their Ljungian counterparts. Unlike the classical identification methods these new techniques enables the inclusion of both structural model errors (bias) and noise (variance) in the estimated uncertainty bounds. However, in order to accomplish this, some prior knowledge of the model error must be available. In general, this prior information is non-trivial to obtain. Fortunately one of these new techniques, denoted the stochastic embedding approach, provides the possibility to estimate, given a parametric structure of certain covariance matrices, the required a priori information from the model residuals. Thus the a priori knowledge is reduced from a quantitative measure. We believe that this makes the stochastic embedding approach superior to the other new techniques for estimating model uncertainty. In this work, new parameterizations of the undermodelling (bias) and the noise are investigated. Currently, the stochastic embedding approach is only well developed for scalar systems. Thus some work is needed to extend it to multivariable systems. This will, however, be beyond the scope of this work. Using the stochastic embedding approach it is possible to estimate a nominal model and frequency domain uncertainty ellipses around this model. It will then be shown how these uncertainty ellipses may be represented or, more correct, approximated with a mixed complex and real perturbation set. This is the link needed to combine the results in robust control and system identification into a step-by-step design philosophy for synthesis of robust control systems for scalar plant, which is the main result presented in this thesis. Throughout the thesis, the presented results will be illustrated by practical design examples. Some of these examples are quite simple, but a few are much more complex. The point of a view taken is that the theories presented should be applicable to practical control systems design. The given examples thus represent a major part of the work behind this thesis. They consequently serve not just as illustrations but introduce many new ideas and should be interesting in their own right.