The impact of robust control techniques motivated the recent development of control oriented identification. This new research area has significantly increased the interest of the community for the `norm bounded' or `set membership' paradigm of uncertainty representation and the related worst case approaches. This is partly due to the deterministic nature of most of the methods for robust control. An additional motivation for this increased interest is the fact that in many cases the `norm bounded' approaches appear to be more suitable to provide reliable bounds for the model error, which is well known to be deterministic rather than stochastic in nature. On the other hand, reliable estimates of accurate models of uncertainty ask for the investigation of advanced robust control design techniques. As an example, consider the case of predictive control which, in spite of its capability of incorporating constraints on the state and control variables, cannot exploit, at least in its classical formulation, the knowledge of bounds on the uncertainty possibly affecting the process nominal model. As an additional example, consider smoothing and filtering theory, which found new motivations in the H_2/H_infinity or set membership theories, with the objective of robustifying classical algorithms or generating new algorithms (multiobjective, central or projection estimators). In this context, the state reconstruction problem with `set membership' disturbances is becoming an important issue. In fact, an efficient solution of this problem would allow for the extension of well-known robust control state feedback techniques to systems where only partial information corrupted by noise is available (output feedback). These recent research advances on robustness of uncertain systems will make it possible to tackle real problems in new challenging applied areas of research. The workplan of the present project reflects this fact: on one side, it aims at a better and more complete integration of the modelling and identification stage with the control law synthesis phase in the design process; on the other side, it intends to propose a new approach to meaningful applications, which may show the advantages and the limitations of the new developments. The first phase of the project is aimed to develop new methods and integrate the existing ones, which show a high level of complementarity and sinergy. As an example, consider the case of invariant regions, successfully investigated by the Udine research group, set membership theory, investigated since long time by the reserch teams of Torino, Firenze and Siena, and predictive control, studied by researchers in Firenze and Siena. Although these three topics, and relative research groups, show significant synergic potentialities for tackling a real problem, they had never been integrated in the framework of a finalized project. In the following, a detailed description of the project research topics is reported in some detail, outlining their reciprocal connections.
ROBUST CONTROL WITH STRUCTURED UNCERTAINTY
Robust control of systems with structured uncertainties, both parametric and norm bounded (H_infinity or l_1), has become a well established research field in recent years. Nevertheless, two major problems still remain unsolved. The first one is related to the complexity of the robust controller. Actually, all the developed techniques generally provide high order controllers. This is one of the main reasons why these techniques have been seldom employed in industrial applications. In fact, it is well known that industrial controllers must have simple structure. This is a very important requirement, especially when a simple implementation permits an easy interpretation of manual or automatic tuning of the controller parameters. The second problem refers to the difficulty of incorporating input or state constraints in the robust control design procedures. Indeed, this is a standard requirement in real world applications. Although mu-analysis techniques provide tools for including these constraints in the formulation of robust control problems (at least in principle), they often result in conservative representations of uncertainty which may dramatically affect the performance of the resulting controller. On the other hand, predictive control techniques allow one to handle hard constraints in a systematic manner, as they are based on a "computational" approach to the control problem. However, robustness of predictive controllers against unmodeled dynamics and persistent disturbances has not been studied in detail and deserves deeper investigations. The two main research lines that are addressed in this project refer to the two open problems described above. In the following, the specific research topics of interest are outlined.
Robust control of uncertain linear systems. The objective of this study is to develop advanced robust control techniques, and to refine and integrate existing control techniques (like gain scheduling or stable dynamic inversion) in order to improve their robustness. The following research topics will be investigated:
a) synthesis of reduced-complexity robust controllers;
b) gain scheduling control for discrete time-varying systems;
c) dynamic inversion of uncertain nonminimum-phase systems;
d) robust control of multivariable systems with structured uncertainties;
e) analysis of periodic controllers for robust control.
a) This topic is strictly related to identification of restricted complexity models. The aim is to develop algorithms for the synthesis of fixed-structure robust controllers, for the case when plants are affected by l_2- or l_infinity-bounded parametric perturbations and H_infinity-bounded unstructured `coprime factor' perturbations. As far as the l_2 case is concerned, when the controller order is unconstrained, the problem boils down to an infinite dimensional convex optimization program. However, when the controller order or its structure are fixed a priori, convexity is lost and the problem becomes extremely complicated. The main objective of this research is to find procedures providing suboptimal controllers, based on the solution of a sequence of convex optimization problems. In this respect, LMI techniques recently developed in the literature seem to be particularly promising. Regarding the l_infinity-bounded perturbation case, interval analysis techniques integrated with branch and bound procedures allow one to infer quantitative information on the set of controllers guaranteeing robust stability and performance of the closed loop system.
b) The second topic, strictly related to robust filtering of time-varying systems, concerns the characterization of problems in which gain scheduling control can be replaced by state feedback robust control. In fact, it is known that for some classes of continuous-time systems, if a stabilizing gain scheduling control exists, then there exists a stabilizing robust control with pure state feedback. However, this equivalence property does not hold for discrete-time systems. For this reason, it is important to further investigate the discrete-time stabilization problem by studying new techniques for the gain-scheduling control synthesis, that take advantage from the on-line parameter measurements to improve system performance. This will require the development of procedures based on the construction of polyhedral invariant sets. In particular, with respect to the activity of the past, the project has the specific goals of the gain scheduling state feedback and output feedback synthesis problems (the latter observer-based) by means of polyhedral Lyapunov functions.
c) This topic concerns the dynamic inversion of uncertain nonminimum-phase SISO systems. Using the results developed on approximate dynamic inversion for systems affected by parametric uncertainties, the joint design of a noncausal reference signal and a feedback controller is pursued, in order to minimize the worst-case settling time subject to the following constraints: closed-loop robust stability; exact steady-state regulation (the internal model principle must be satisfied); fixed overshoot and undershoot of the output; saturation bounds on the control variable (avoidance of the wind-up phenomenon).
d) This research topic is mainly devoted to determining robust solutions for tracking and regulation problems under structured uncertainties of the plant. Here exogenous signals are assumed to belong to known classes, possibly under additional requirements, such as the nominal input-output decoupling, for continuous-time, discrete-time and single- or multi-rate sampled data systems (with the further requirement of ripple-free error response in the last case). Moreover, the structure of perturbations for which well-known conditions guarantee robust stability with a given compensator will be studied, with the aim of extending such conditions to allow for inclusion of wider classes of plants.
e) This investigation is devoted to deepen the study of the advantages deriving from the use of periodic controllers for the robust control of linear time-invariant discrete-time plants with highly structured uncertainties, such as unknown different scalar gain factors acting on every scalar input/output of the plant. Here, the aim is to reach robust stabilization with large, possibly unbounded, gain margin, and robust regulation and tracking for exogenous signals belonging to known classes.
Robust predictive control. In this research field, the project will address open problems related to robustness against unmodeled dynamics and persistent disturbances. The following topics will be investigated:
a) algorithms for robust predictive control;
b) predictive control based on `set membership' state observers;
c) stabilization of linear systems with saturations.a) With reference to the first topic, this research aims at studying robust predictive control techniques capable of facing various types of uncertainty. A promising approach that will be pursued is based on the theory of robust invariant sets. In order to guarantee robustness with respect to various sources of uncertainty, the key idea is to force the system state to remain in a suitable robust invariant set, to be determined off-line by means of set computation techniques, on the basis of a-priori knowledge about the class of admissible disturbances and/or models. Another open problem related to predictive control, is enhancing the controller performance when the system is affected by persistent disturbances. In this case, constraint fulfillment and performance optimization usually lead to the formulation of complex min-max problems. Since the control action is based on open-loop prediction of the system evolution, the obtained controllers are often very conservative. For this reason, predictive control schemes based on closed-loop predictions will be investigated.
b) In the case of partial state information (output feedback), the presence of unknown-but-bounded disturbances can be tackled by set membership state observers, which provide the set of states compatible with output measurements and disturbance bounds. The objective is to combine model predictive control and set membership state estimation in order to guarantee that at each time instant the input and state constraints are satisfied for all the feasible states and disturbances within the given bounds.
c) It is well known that, in the disturbance-free case, linear systems with input saturations are controllable to zero from an arbitrary initial state and with an arbitrarily small bounded input, provided that they are not exponentially unstable. If a persistent disturbance is acting on the plant, this objective must be obviously modified: in particular, the question is whether for such systems it is possible to drive any initial state to a sufficiently small neighborhood of the origin (robust invariant set), provided that the input bounds are sufficiently large compared to the disturbance bounds. If this objective is feasible, the aim is to develop robust predictive control techniques which guarantee its fulfillment.ROBUST CONTROL ORIENTED IDENTIFICATION
Worst-case optimality criteria and representation of uncertainty through norm-bounded operators, are among the main features of this recent research field and represent unifying aspects for the activity of the research units involved in this project.
Identification of restricted complexity models. The main objective is to find optimal and suboptimal worst-case estimators for restricted complexity models (see the Invited Session `Identification for Control: Consistency, Complexity and Optimality', 36th IEEE Conference on Decision and Control, San Diego (USA), December 1997, organized by the coordinator of this proposal). This problem is particularly important since it is well known that complex models require more involved estimation procedures and make control design much more difficult, when H_infinity or l_1 techniques are adopted. In the Information Based Complexity setting, the optimal solution of this problem leads to the definition of conditional central estimators. In general, these estimators require the solution of involved min-max optimization problems. The results pursued in this project include: characterizing the conditional center as a function of the norm bound assumed for the error; finding classes of suboptimal estimators that exhibit a lower computational complexity, in spite of an estimation error larger than the radius of information, i.e. the minimum achievable error. In this context, the computation of guaranteed and nonconservative upper bounds on the estimation error provided by suboptimal estimators appears an interesting issue. Moreover, the properties of some classes of projection algorithms, that allow for the solution of the restricted complexity model estimation problem through linear programming techniques, will be investigated. One of the main features of these estimators is the possibility of including several uncertainty sources (e.g., different types of nonlinearities) in the considered model structure. A further objective within this research theme is to devise new methods and linear programming based algorithms to deal with the problem of a priori information validation. In fact, the presently available techniques suffer from excessive computational effort required with increasing number of available measurement data. Finally, restricted complexity models will be studied in the context of the `identification-control interplay'. This interplay will be studied in two different aspects. The first one regards the use of a suitable metric for the identification error, in order to correctly evaluate the deterioration of the performance index used for the controller design. This deterioration is obviously due to the lack of knowledge of the true plant. The second aspect consists in the estimation of bounds on the performance guaranteed by a controller designed on the basis of an assigned complexity uncertainty model.
Identification of complex systems. Recent techniques for identification of complex systems (i.e., systems that cannot be adequately approximated by finetely parameterized models) will be investigated. The main purpose is to show analogies and differences between these techniques and the conditional approach addressed in the previous research topic, in order to find new estimation algorithms that can handle a set membership description of uncertainty in the identification of complex systems. An important objective of the project is to investigate set membership identification techniques for uncertainty models in which complexity is accounted for by nonlinear terms. As a first step towards the extension of set membership identification metodologies to nonlinear systems, special classes of nonlinear plants made up by a static nonlinear block suitably connected to a linear system will be considered. In particular, Wiener, Hammerstein and Lurie systems will be studied. Set membership identification techniques allows one to identify not only a single nominal model, as performed by classical identification metodologies, but a set of models (called "uncertainty model"), which is able to account for uncertainties due to finite noisy data and partial a priori information. For the class of nonlinear systems described above, a set membership identification procedure will be developed. In particular, the results of the identification will be an uncertainty model made up by a nominal model, with an additive dynamical perturbation accounting for the unmodeled dynamics, while the uncertainties on the nonlinear part will be characterized by bounded sectors. The above uncertainty representation is well suited for robust control design. The last topic which will be investigated regards identification of Linear Parameter Varying (LPV) models. These models have been recently proposed for the analysis and design of gain scheduling controllers. The use of LPV systems is potentially interesting as it gives the possibility of using robust control techniques for gain scheduling design. In fact, in such systems the time variation is not modelled dynamically, but it can be dealt with as an exogeneous variable. In this context, the aim is to devise set membership based identification schemes accounting for both model uncertainty and noise present in the control system. A model validation approach will be used to generate robust control oriented techniques.