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Siena Research Unit

The research project is focused on the joint development of identification techniques and modern robust control methodologies, for dynamic systems. The specific task of the Siena Research Unit is to study possible uncertainty representation techniques, leading to mathematical models that are suitable for identification and robust control of uncertain dynamic systems. Recent years have witnessed a remarkable growth of interest for this research field, especially for what concerns identification for control (see the special issues [1] and [2] and many invited and contributed sessions in recent IEEE Conferences on Decision and Control and IFAC Symposia on Identification and System Parameter Estimation). The two main classes of uncertainty representation techniques are usually addressed as `soft bound' and `hard bound' methods. The former are based on stochastic models of the uncertainty and/or the system, while the latter assume that the uncertain variables satisfy deterministic assumptions that can be described through suitably defined norm bounds. In this project, methodologies and algorithms will be developed in the `hard bound' setting. In this context, two possible uncertainty representations are considered in most contributions: `set membership' and `norm bounded' (H_infinity, H_2, L_1). Both representations will be addressed in this project, using the Information Based Complexity theory as a unifying framework for formulating problems, studying optimality of the proposed algorithms and providing critical interpretations of the optimality criteria adopted in the `set membership' and `norm bounded' settings. The four research topics to be developed refer to the case when parametric uncertainties, process disturbances, and measurements errors are bounded in maximum (l_infinity norm) or mean square (l_2 norm) sense.

Identification for robust control. This research aims at developing recursive and batch identification algorithms in the `set membership' setting. The identified models must be suitable for the most popular robust control strategies for uncertain sytems. Two main subjects will be investigated:
a) estimate of restricted complexity models;
b) identification of complex systems.

a) With reference to the first topic, 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). This problem is particularly important since it is well known that complex models require more complicated 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 complicated 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 algorithms is particularly interesting.
b) With reference to the second topic, some recent techniques for the identification of complex systems (i.e., systems that cannot be adequately approximated by finetely parameterized models [3]) will be investigated. The main purpose is to show analogies and differences between these techniques and the conditional approach described in a), in order to find new estimators that can handle a set membership description of uncertainty in the identification of complex systems.

Optimal state estimation and filtering. The objective here is to develop optimal and suboptimal algorithms for state filtering and smoothing, in the set membership approach founded on the Information Based Complexity theory. Classes of optimal and suboptimal estimators, like conditional central algorithms and interpolatory algorithms, deserve particular attention as they provide new estimation tools in this context. Moreover, a comparison with well-known estimators, like the classical Kalman filter or the H_2 and H_infinity filters [4-5], minimizing the norm of the estimation error operator, is of great theoretical and practical interest, especially in view of the different hypothesis under which these estimators are derived.

Synthesis of reduced-complexity robust controllers. This topic is strictly related to subject 1). The aim is to develop algorithms for the synthesis of fixed-structure robust controllers, for the case when plants are affected by l_2-bounded parametric perturbations and H_infinity-bounded unstructured `coprime factor' perturbations. When the controller order is unconstrained, this 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 [7] seem to be particularly promising.

Robust predictive control. Model predictive control has become the accepted standard for multivariable constrained control in the process industry [8]. The control policy consists of solving at each sampling time, starting at the current state, an open-loop optimal control problem over a finite horizon. At the next time step the computation is repeated starting from the new state and over a shifted horizon. The solution relies on a dynamic model of the process, and is able to enforce input and output constraints, while optimizing a quadratic performance index. The most important open problems in predictive control concern robustness against perturbations of the nominal model and/or persistent disturbances. The main topics that will be investigated in this research project are: a) algorithms for robust predictive control; b) predictive control based on set membership state observers. a) With reference to the first topic, it is planned to investigate robust predictive control schemes where a nominal performance index is minimized and hard state and input constraints must be satisfied. In order to guarantee robustness, the idea is to impose that the system state belongs to a robust invariant set [9]. When the system is affected by persistent disturbances, constraint fulfillment and performance optimization usually lead to the formulation of complex min-max problems. Since the control action is based on the 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 [10] will be investigated. Preliminary studies were presented in [11]. 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 for all the disturbances within the given bounds. Preliminary results were obtained in [12].

Project organization. With reference to the four topics described above, the project is divided into two phases that will be developed during the first and second year of the project, respectively. In the first phase, the methodological aspects of all the topics will be addressed. The second phase of the project will be devoted to algorithm implementation and testing, and to the application of some of the developed techniques to specific problems. More precisely, the activity will focus on:
1) algorithms and procedures for the computation of conditional centers;
2) recursive algorithms for set membership state smoothing and filtering (in collaboration with the Pavia research unit);
3) LMI optimization procedures for the synthesis of robust controllers (in collaboration with the Firenze research unit);
4) constrained optimization algorithms for robust predictive control (in collaboration with the Firenze and Udine research units).
Algorithms relative to points 1) and 2) will be employed in estimation and filtering problems for dynamic vision and the construction of navigation maps for autonomous mobile robots. Moreover, the estimation techniques developed in the project will be applied to the identification of complex biological and environmental systems. Preliminary work along these research lines have been developed in [13,14].

Interactions with other research groups. As far as the role of the Siena Research Unit in the national project is concerned, topic 1) is mostly developed by the research group in Siena and will be enriched and completed by the results obtained by the units of Torino and Palermo. With reference to topic 2), the unit of Pavia will provide important knowledge concerning robustness and performances of adaptive state estimators. Topic 3), which is investigated in collaboration with researchers of the Firenze unit, is strictly connected to the research on robust control carried out by the units of Napoli and Parma. Finally, the Udine research unit will provide a useful background on the use of invariant sets for control purposes. The results obtained in the study of predictive control techniques will be completed by the parallel research on this topic, developed by the units of Firenze and Torino.

Publications.

Roma Research Unit

The unit research activity has been focused on the following four main themes:
1) control of linear time-invariant systems;
2) control of sample-data linear systems;
3) control of linear periodically time-varying systems;
4) periodic control of linear time-invariant systems.

Control of time-invariant linear systems. A first set of research products obtained on this subject is concerned with a comparison between pairs of robust stability conditions proposed in the literature to deal with unstructured perturbations. The comparison is performed by comparing the set of perturbed systems whose stability can be guaranteed by each considered robust stability condition once the compensator is fixed. Specific situations have been pointed out in which the use of one condition is preferable over the use of the other one, since the former guarantees robust stability for a wider class of perturbed systems [C1]. A second set of research products obtained on this subject is again concerned with robust stability in the presence of unstructured uncertainty. Two sufficient conditions based on the H_infinity norm have been derived by which robust stability and performance can be guaranteed. Taken together, the proposed conditions generalize some already known conditions; one of the two conditions is preferable over other existing conditions since it is amenable to be used in relation with a wide range of performance requirements [C11]. A third set of research products obtained on this subject is concerned with anti-windup control for systems affected by relevant input saturation effects. In particular, anti-windup techniques are aimed at designing compensators which are not only able to guarantee a satisfying closed loop behaviour when the saturation activates, but also to recover, at least for sufficiently small initial conditions and exogenous inputs, the behaviour induced by an a-priori fixed compensator designed for the system without saturation. The design technique proposed in [C5] is based on the use of linear matrix inequalities (LMI), and is then particularly attractive due to the large number of efficient algorithms available to solve such numerical optimization problems. Interesting experimental results for a mechanical system have been shown in [C10]. Additional research on this technique has lead to advanced anti-windup synthesis results, characterized by robustness properties [C12-C13] which can possibly be pre-assigned [C13]. A fourth set of research products obtained on this subject is concerned with linear mechanical systems subject to impacts [R4], [R5], [C4], [C8]. For such a class of systems, and for the case of ``non-smooth'' impacts, a dead-beat control law has been proposed which is able to stabilize a contact configuration [C4] while avoiding the arise of limit cycles experienced when simpler, naiver control laws are adopted. The main result in [C8] is the generalization of the Youla-Kucera parametrization to the case of linear mechanical systems subject to impacts; the approach has been applied to a physically motivated example for which the combined used of the proposed parametrization and the internal model principle has made possible the solution of a problem of robust regulation to a contact configuration in spite of relevant uncertainties on the values of the physical parameters of the system.

Control of sample-data linear systems. Problems concerning the asymptotic tracking of references belonging to known classes and the asymptotic rejection of disturbances belonging to known classes, robustly with respect to uncertainties in the description of the controlled plant, have been addressed and solved, under the additional constraint of "ripple-free" convergence of the error response, both in the case of single-rate sampling and in the case of multi-rate sampling, and both for uncertain "physical" parameters of the plant and for independent variations of the elements of the matrices characterizing the controlled plant. About such problems, necessary and sufficient conditions for the existence of a solution have been stated in [R2], and compensators able to solve the same problems have been proposed. In particular, a complete continuous-time internal model of the exogenous signals has been proven to be necessary for the solution of the above mentioned problems, even in the case of multi-rate compensators and when the required convergence to zero of the free response and the error response of the closed loop system is of an exponential kind (whereas previous conditions only dealt with the case of dead-beat convergence). A second kind of contribution whose solution is closely related to the use of a discrete-time model of the considered system is concerned with the control of mechanical linear systems subject to inequality constraints (whose purpose is to model possible impacts). A class of such systems as been considered in [C7], characterized by the property of being not controllable in absence of impacts; it has been shown in [C7] that, exploiting the impacts, it is possible to obtain regulation to zero of the uncontrollable part of the system via a hybrid control algorithm (i.e. a control algorithm involving both a continuous-time part and a discrete-time part).

Control of periodic linear systems. The problem of obtaining a periodic state space representation of a process described by linear difference equations with periodic coefficients has been considered; clearly, such transformation is a necessary preliminary step in order to design a controller for the system under study. Previously developed theory provided a solution to the above mentioned problem by using the concept of system equivalence, but only under the assumption that the process had no null characteristic multiplier. Such an assumption is quite strong, since it excludes important classes of periodic systems, e.g. the discrete-time periodic models of multi-rate control systems. Such an assumption has been removed in [R1], where necessary and sufficient conditions for the existence of a state space discrete-time model have been proposed. The proposed conditions can be checked directly on the original description of the process. The strict relation between the structural properties of the obtained state space model and the structural properties of the given process has also been proven. Moreover, the analysis of a suitable periodic discrete-time error system derived by sampling the real, continuous-time error system, has been used to prove the result in [R3], concerned with a problem of periodic trajectory tracking for a mechanical system subject to non-smooth impacts (such a system is called a "billiard" in the mathematical literature).

Periodic control of linear time-invariant systems. About the use of linear periodic controllers for linear time-invariant discrete-time systems, the problem of stabilization with infinite gain margin (with respect to possibly different unknown scalar gains on each scalar input) has been studied and solved in [C2], also guaranteeing the additional requirements: a) to obtain an asymptotically stable compensator, b) to obtain that the closed loop stability for which the infinite gain margin has been achieved is such that it guarantees an a priori fixed exponential rate of convergence of the free responses of the closed loop system, regardless of the values of the unknown scalar gains. For such a problem, necessary and sufficient conditions for the existence of a solution, and an algorithm to design a suitable compensator able to solve the problem, have been given in [C2]. The same kind of problem has been studied in [C3], substituting the requirement of asymptotic stability of the compensator with the requirement of asymptotic tracking of references belonging to known classes. Also in this case it has been possible to give conditions under which the problem is solvable, and a design procedure for a compensator satisfying the requirements of the problem. Finally, the same kind of problem already solved in [C3] has been considered in [C6], but for the case when the unknown scalar gains (with respect to which the gain margin is defined) affect the output (and not the input, as before) of the controlled plant. Also for such a problem it has been possible to give the necessary and sufficeint conditions for the existence of a solution, and a synthesis procedure for the compensator; quite surprisingly, the controller thus obtained has a structure which is much simpler than the structure of the controllers proposed in the previous papers [C2] and [C3].

Publications.

Udine Research Unit

The unit research activity has been focused on control synthesis techniques based on invariant sets approach. Such approach has recently received a renewed interest due to the capability of providing exhaustive solutions to challenging synthesis problems, also thanks to the improved computer performances, which allow for the actual implementation of the algorithms derived thru such approach to problems whose numerical complexity appeared intractable a few years ago. The main research topics carried out by the unit have been the control via gain scheduling techniques, robust adaptive control, model predictive suboptimal control and the synthesis of control laws for production-distribution systems.


Gain scheduling. The considered gain scheduling techniques, differently from the robust ones, aim at exploiting on line the knowledge of the structured uncertainty affecting the system for deriving stabilizing control laws.

For specific classes of problems, in the continuous time case [1], it was shown that the robust and the gain scheduling stabilization problems are somehow equivalent, in that the stabilizability conditions for the gain scheduling problem, say when the parameter is known at every time instant, are the same for the stabilization of the same uncertain system when the uncertain parameter value is not available for control purposes. For the same class of systems, in the discrete-time case, things are not quite the same, and robust and gain scheduling stabilizability conditions turn out to be completely different.

The unit has analyzed such differences [2], and for the discrete-time case a duality theory has been developed which shows that the gain scheduling linear state feedback stabilization problem and the gain scheduling linear state estimation problem are dual to each other.

Moreover, it has been shown that linear gain scheduling state feedback stabilizing control laws cannot be outperformed by nonlinear gain scheduling stabilizing control laws.


Invariant sets/Application. The study of the class of initial condition starting from which constant reference signals can tracked for linear systems with input and state constraints has been investigated, exploiting the theory of invariant sets, in [4]. It has been shown that the set of initial conditions starting from which the state evolution can be steered to the origin in the presence of input and state constraints can be successfully and efficiently used to derive state feedback stabilizing control laws for tracking purposes.

The application of the results based on robust invariant sets theory for the derivation of a stabilizing controller for a compressor surge has been carried out in [5,6]. The derived adaptive controller has been successfully implemented on a lab compressor surge system.

The study of an industrial device, a rotating channel used in a steel caster, has been carried out in [13]. Using classical control techniques, a dual mode controller has been derived which allows for the satisfaction of the stringent performance specifications imposed by the industry while guaranteeing a good level of robustness wrt the uncertain parameters affecting the system.


Suboptimal/Receding horizon. Receding horizon techniques were in the past usually exploited to derive optimal (or suboptimal) stabilizing control laws for dynamic system with slow dynamics. The basic problem limiting the use of such techniques, which are essentially based on on-line optimization, was the high number of free variables over which the minimization had to be carried out.

The authors have proposed a discretization scheme [7], based on the Euler discretization, which allows to derive suboptimal receding-horizon control laws without the need to resort to time-consuming on-line optimizations. It has been shown that known receding horizon techniques can be successfully used even for the control of continuous-time systems, provided the continuous-discrete time discretization is the Euler one, a property which doesn't hold in general if the classical exponential (for linear systems) discretization is used.

The use of an Euler discretization allows for the derivation of simple control laws which can be implemented also with systems with fast dynamics and very small sampling time.


Production. This research aims at combining invariant set theory with structure-specific control problems, to derive simple and efficient control laws for production-distribution systems.

Exploiting the structure of the considered class of systems, the researchers of this unit have been able to solve challenging problems such as the stabilization of uncertain production-distribution systems in the presence of unknown-but-bounded disturbances and setups and time optimal worst case control problems [11,10,9,12,8].

Publications.

 

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