Identification and Robust Control

Modeling and identification procedures of real processes should be designed having in mind their specific purposes. Although identification techniques have always taken into account typical requirements of control tasks [1], recent improvements in high performance robust and adaptive control theory ask for the study of special identification procedures tailored to these advanced control techniques. Research in robust control assumes that uncertainty bounds, like tolerance bands on frequency response (H_infinity limits), are known. However, from the identification viewpoint, little effort had been devoted to the estimation of such uncertainty models until a few years ago. Motivated by the advances in high performance robust control, the research communities in identification and robust control have started a synergetic work by stressing their similarities and complementarities (see, e.g., Special Issues [2-4]). In the robust control field a worst-case approach is usually adopted which guarantees the desired performance for the worst-case uncertainty realization [5,6]. As far as control oriented identification is concerned, problems are dealt with in the literature according to two approaches. In the first approach, uncertainty is handled as a stochastic quantity (soft bounds), while in the second one it is treated as a deterministic variable (hard bounds).
Research by the Siena Systems and Control Group in this field mainly refers to the hard bound approach, where disturbances and uncertainty on nominal models are assumed to be bounded in some norm [7-10]. A summary of recent results achieved by the Siena Systems and Control Group, along two main research lines, is reported below.

ROBUST CONTROL ORIENTED IDENTIFICATION.
As pointed out in [7-11] and in the survey papers [12-13], recent research progress in this field involves complexity, optimality and robustness of pointwise and set estimators, both for identification and filtering; optimal design of experiments; interplay between identification and control (see for instance the invited sessions [11,14,15], the book [10], and the Special Issue [4]). The scientific expertise and the research activities recently developed by the Siena research group can be outlined as follows.
a) Set membership identification and state estimation. Recent research activities deal with the design of optimal recursive and batch set estimators for both parameter and state estimation. In particular, several algorithms based on parallelotopic approximations, for the recursive estimation of the feasible parameter set [12,16,17] and the feasible state set [18], were developed.
b) H_2, H_infinity, and l_1 identification. Several contributions were given in the field of optimal and suboptimal pointwise estimators, like projection, interpolatory and central estimators. The common research effort yielded a mixed parametric/non-parametric approach, based on the identification of a nominal model within a given set of finite-dimensional parametric models and the estimation of an upper bound on the model uncertainty, measured by H_2, H_infinity, or l_1 norms [13,19-21]. In this context, important results were obtained on the identification of restricted complexity models for real processes, which is considered a key topic in identification [23-24] since long time [22]. More recently, the problems of optimal orthonormal basis selection and optimal input selection with respect to the worst-case error provided by central projection estimators have been investigated [25,26].
c) Set membership identification of nonlinear models. Motivated by recent developments in hybrid system theory and applications, a new research line focused on identification of nonlinear structures via piecewise affine models has been started. The approaches in [27-29] are based on randomized algorithms or integer programming and yield remarkable performance in terms of both model quality and computational effort. Active research is currently carried out by the Siena research unit on the application of the estimation and identification techniques introduced above, including set membership estimation and filtering for dynamic vision, localization and navigation maps in autonomous mobile robotics [30-33]. Finally, a software application for web-based remote identification experiments on real processes has been developed (http://act.dii.unisi.it)[34].

ROBUST CONTROL WITH STRUCTURED UNCERTAINTY.
The main feature of synthesis methods for unstructured uncertainties is that they allow one to find closed-form solutions to control synthesis problems. The inherent limit of these approaches is that they may be quite conservative in practical applications. Research on techniques and algorithms for problems with structured uncertainties is quite recent. The mu-analysis/synthesis [6] and the parametric robustness analysis [5,35] are among the most popular approaches to solve a robust control problem with structured uncertainty. Recent results on robust control with structured uncertainties by the Siena research, are described below.
a) Stability and robust performance analysis with parametric and unstructured uncertainty.
Contributions were provided on stability and robust performance for linear systems affected by parametric perturbations linearly correlated, polynomially correlated, or mixed with unstructured (H_infinity) uncertainties [35-37]. An extensive bibliography on this subject is reported in the survey paper [35].
b) Optimization techniques for control systems with structured uncertainty. The main drawback of analysis and synthesis methods for systems with structured uncertainty is the high computational complexity. In this context, by exploiting convex Linear Matrix Inequality (LMI) optimization techniques [38], a novel approach has been introduced for solving minimum distance problems from polynomial surfaces [39]. This has allowed for the development of efficient methods for solving complex problems such as the computation of the parametric stability margin of control systems with structured uncertainty [39] or the synthesis of robust controllers with a fixed structure, e.g. PID or lead-lag, for systems with parametric perturbations [40-42]. An interesting development of the above convex optimization techniques is the construction of polynomial Lyapunov functions, which are aimed at reducing the conservatism introduced by traditional quadratic functions. By combining homogeneous polynomial forms and LMI optimization, the problem of computing polynomial homogeneous Lyapunov functions for systems with structured time-varying uncertainty has been addressed with promising results [43,44].
c) Robust model predictive control (MPC). In MPC [45-46], at each sampling time an open-loop optimal control problem is solved over a finite horizon. For systems affected by uncertainty, a typical strategy consists of solving a min-max optimization problem, which however turns out to be computationally very demanding for on-line implementation. In order to push MPC to fast-sampling applications (automotive, mechanical, aerospace, etc.), techniques based on multiparametric programming were recently developed to find the equivalent piecewise affine state-feedback form of the MPC control law, that has tremendous computational advantages over standard on-line optimization solvers [47-50].
The book [10], the workshop 'Robustness in Identification and Control', which was held in Siena in 1998, the Special Issue on 'Robustness in Identification and Control' [4] organized and edited by researchers of Siena, and the Matlab toolbox 'Model Predictive Control' co-authored by some researchers of Siena testify the effort and the lively interest in this area of research.

References

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