Special issue on ‘periodic systems and robust control’ dedicated to Osvaldo Maria Grasselli

Abstract

This special issue celebrates the career of Osvaldo Maria Grasselli. The idea of putting together a special issue dedicated to Osvaldo surfaced during the workshop ‘One-Day Symposium on Advances and Challenges in Linear Control Systems’ organised at the ‘Università di Roma ‘Tor Vergata’ on March 2012, on the occasion of his 70th birthday. This special issue, therefore, contains a record of the technical programme. The research vision of Osvaldo is conspicuous in all contributions, spanning the areas of robust control, geometric control, control of multivariable systems, periodic control systems and their applications.

1. Introduction

Osvaldo’s career has spanned four decades, during which he has equally contributed to the development of new ideas and to the education of whole generations of control engineers and PhD students, in various Italian universities.

Osvaldo was born on 1942 in Rome (Italy). He received his Laurea (M.Sc.) degree in Electrical Engineering in 1968 from the ‘Università di Roma’ (now known as ‘La Sapienza’). Soon after graduation, he began his research activity in the Istituto di Automatica under the direction of Professor Antonio Ruberti, in the area of control system analysis and design.

In 1971 he was appointed Ricercatorein the Ugo Bordoni Foundation, and in 1972 he moved to the National Research Council (C.N.R.) while continuing his research activity at the Istituto di Automatica. In 1973, upon invitation of Professor H.H. Rosenbrock, he joined as a visiting scholar the Control Systems Center of the University of Manchester Institute of Science and Technology. In the same year, he became Professore Incaricatoat the ‘Università di Ancona’ (now the ‘Università Politecnica delle Marche’), where he initiated the course Systems Theory and then, from 1976, he was appointed Director of Studies for the Electrical Engineering Degree. As evidence of his early commitment to education, during that period he wrote two graduate-level textbooks (Grasselli, 1978; Grasselli & Leo, 1979). At the same time he gave significant research contributions, some of which in collaboration with Alberto Isidori and Fernando Nicolò, on the analysis and design for linear time-invariant systems and bilinear systems.

In 1981 he was appointed Professore Ordinarioof Systems Theory at the Faculty of Engineering of the ‘Università di Ancona’ while retaining his position as Director of Studies. At the same time, he started a long-term research partnership with Sauro Longhi directed at unveiling the properties of linear periodic systems. In 1984 he moved to the Faculty of Engineering of the ‘Università di Roma ‘Tor Vergata’, as Professore Ordinarioof Control Theory, and was subsequently appointed Director of Studies for the Electrical Engineering Degree (1985–1987).

Following his early work on linear periodic systems, he devoted the next decade to the development of analysis and design methods for linear periodic discrete-time systems. Some of his major contributions, in cooperation with Sauro Longhi, are the characterisation of the zeros, the design of pole placement algorithms and the geometric theory of periodic systems. In addition, together with Sauro Longhi and Antonio Tornambè, he developed a periodic-system counterpart of Rosenbrock realisation theory, thus establishing a natural method to represent general linear periodic models in state-space form, and a robust regulator theory for linear time-invariant plants with parametric uncertainties. Finally, he pioneered the study of multi-rate sampled-data systems, creating an active research group currently comprising Paolo Valigi, Laura Menini, and Sergio Galeani.

He has taught for almost three decades a course on Multivariable Control and, for over a decade, courses on Robust Control and Systems Theory. This extensive teaching experience has resulted in the publication of three graduate-level textbooks (Grasselli & Galeani, 2012, 2013; Grasselli, Menini, & Galeani, 2008) and in the appointment to the position of Director of Studies for the Degree in Automation Engineering (2001–2005).

2. Bilinear systems

Osvaldo Maria Grasselli has pioneered, in collaboration with Alberto Isidori and Ferando Nicolò, the study of bilinear systems. In particular, he solved the stabilisation problem for discrete-time bilinear systems with independent additive and multiplicative control (Grasselli, Isidori, & Nicolò, 1980) and established the connection between the controllability of the system and the existence of a dead-beat controller with a two-layer structure. In addition, he derived conditions for controllability of a special class of homogeneous bilinear systems together with conditions for the solution of the output regulation problem (Grasselli, Isidori, & Nicolò, 1979).

He continued to develop this research area with his first PhD student, Sauro Longhi, during his period at the ‘Università di Ancona’. In particular, they studied the stabilisation problem for continuous-time bilinear systems showing that the stabilisability of particular linear or homogeneous bilinear subsystems is sufficient for stabilisability of the underlying bilinear system (Grasselli & Longhi, 1983).

3. Periodic systems

Motivated by the study of bilinear systems, Osvaldo moved to study analysis and design problems for periodic systems. He showed that periodic controllers may be suitable for stabilising bilinear systems and designed a class of dead-beat controllers for linear periodic discrete-time systems achieving output regulation for periodic disturbances (Grasselli & Lampariello, 1981). These early contributions were followed by the study of canonical decompositions for linear periodic systems, which generalise the Kalman canonical decomposition (Grasselli, 1984). Notably, the same problem was studied in the same period by Bittanti and Bolzern (1985). Recently, the problem has attracted new interest, and the study of canonical decompositions with variabledimensions of the various subsystems has been initiated. This issue was raised in the early paper (Grasselli & Longhi, 1991c), whereas explicitly algorithms have only recently been developed by Varga (2004) and Verriest (2004).

The application of these decomposition algorithms in the design of dead-beat controllers and functional dead-beat observers for linear periodic systems has been reported in Grasselli and Longhi (1986b) on which they rely the results of Grasselli and Longhi (1991c) and Longhi and Zulli (1996).

4. Geometric methods

The central role of structural properties in Osvaldo’s research path was instrumental in the introduction of the geometric approach for the study of periodic systems, thus extending the classical results of Basile and Marro (1992) and Wonham (1979). In particular, the notions of controlled invariant subspaces (Grasselli & Longhi, 1986a), conditioned invariant subspaces (Grasselli & Longhi, 1987), and inner and outer controllable subspaces were introduced. The latter, in particular, proved instrumental to solve the disturbance localisation problem (Grasselli & Longhi, 1986a). Similarly, the dual problem, namely the design of linear functional observers with disturbance localisation, was solved exploiting the notion of outer reconstructible subspaces (Grasselli & Longhi, 1987). Finally, these notions were exploited in Grasselli and Longhi (1988a) to solve the disturbance localisation problem by measurement feedback.

The papers (Grasselli & Longhi, 1991b, 1993) provide surveys of these notions. In addition, the former offers a new time-invariant geometric characterisation, which lends itself to the development of novel algorithms for the computation of supremal and infimal subspaces. The latter, instead, introduces the notion of inner reachable subspaces which, together with the pole placement algorithm in Grasselli and Longhi (1991c), yields the solution of the non-interacting control problem via static state feedback.

5. From models to systems

The time-invariant representations, exploited by Osvaldo in the analysis and design of periodic systems, and in the characterisation of their pole-zero structure, form the basis for the construction of a periodic state-space description for linear processes modelled by linear difference equations with periodic coefficients.

Notably, the paper (Grasselli, Tornambe, & Longhi, 1994) develops a polynomial time-invariant description of a linear periodic processes, which is the periodic counterpartof the Rosenbrock’s polynomial matrix description for linear time-invariant processes. Systems equivalence and causality conditions were then derived in Grasselli, Longhi, and Tornambe (1995a), Grasselli, Longhi, and Tornambe (1995c) and Grasselli, Longhi, and Tornambe (2001).

6. Multi-rate systems

A special class of periodic systems is the class of multi-rate sampled-data systems. Starting from preliminary results on the selection of the sampling time in Longhi (1994), a series of control problems were studied: in Grasselli, Jetto, and Longhi (1995), under an additional requirement on the existence of a continuous-time internal model, the problem of dead-beat ripple-free tracking for multivariable multi-rate systems was solved. The same perspective was advocated in Grasselli, Longhi, and Tornambe (1994b), Grasselli, Longhi, Tornambe, and Valigi (1996b) and Grasselli, Menini, and Valigi (2002), in which the robust (against variation of physical parameters) ripple-free tracking and disturbance rejection problem were solved, showing the necessity of a continuous-time internal model of the exogenous signals.

7. Robust control

The role of changing parameters in control problems has been a leitmotivin Osvaldo’s research career. The design of robust control laws, achieving at the same time robust regulation and disturbance rejection, for general multivariable linear systems has been presented in Grasselli and Longhi (1991d) and Grasselli, Longhi, and Tornambe (1993d). The applicability of this ideas to the design of robust control laws, achieving asymptotic regulation, for linear periodic systems has been reported in Grasselli, Longhi, Tornambe, and Valigi (1996a) and Grasselli, Menini, and Valigi (1999a).

8. Scanning the issue

The paper by Bianchini, Paoletti, and Vicino (2013) studies the L₂-gain of discrete-time piecewise affine or piecewise polynomial systems by using polynomial and piecewise polynomial storage functions, semidefinite programming and sum-of-squares decomposition. As an application, the PWA model of an electronic component placement process in a pick-and-place machine, identified from experimental data, is considered.

Longhi and Monteriù (2013) solve a fault detection and isolation problem for linear discrete-time periodic systems. The solution, consisting of an observer-based residual generator where each residual is sensitive to one fault and insensitive to the others, is based on geometric tools, and, in particular, on the outer observable subspace notion; an algorithmic procedure for the design is included.

The paper by Varga (2013) surveys the more recent techniques about computational problems for linear periodic systems: numerically reliable and efficient algorithms for manipulating matrix products, reduction of large-scale structured matrix pencils, computing with discrete-time periodic system models with time-varying state dimensions and multiple shooting techniques to solve periodic matrix differential equations. Some of still open computational problems are pointed out.

Corradini, Cristofaro, Orlando, and Pettinari (2013) consider the robust transient shaping of periodic linear discrete-time plants subject to saturating actuators, in the presence of bounded matched uncertainties. A constructive procedure, which employs a time-varying sliding surface to guarantee the ultimate boundedness of the state trajectories, is presented.

In Menini and Tornambe (2013), a normal form for linear periodic discrete-time systems depending on physical parameters is proposed, thus extending a similar notion proposed by Arnold (1971) for matrices (time-invariant autonomous systems), and extended by Tannenbaum (1981) to time-invariant linear control systems (with inputs and outputs). Any linear periodic discrete-time system can be transformed into normal form by a linear periodic change of coordinates depending on the parameters, being identity at the nominal values of them.

The paper of Bolzern and Colaneri (2013) deals with the stability analysis of discrete-time periodic switched linear systems, which consist of a network of time periodic linear subsystems sharing the same state vector and an exogenous switching signal that triggers the jumps between the subsystems. Arbitrary switching signal and signals satisfying a dwell-time constraint are considered. Stability conditions are given in terms of LMI conditions, and guaranteed H₂ and H₁ performances are provided.

Discrete-time linear periodic systems are characterised in Verriest (2013) using cyclic projection operators on sequences, showing that the two well-known liftings of periodic systems to a time-invariant one, the monodromy and the cyclic representations, can be easily obtained using this framework. This approach also allows the definition of the operational transfer function for such systems, and more generally, the operational transfer inclusion.

In Luna, Fierro, Abdallah, and Lewis (2013), a control system approach, which allows the compression of the shared information among mobile agents that take part in multi-vehicle missions, is proposed. The proposed method aims to simplify the calculation of the stationary errors by computing the sign of the errors rather than their exact values and the control law may then be used to stabilise the system.

Flexible structures with collocated force actuators and position sensors lead to negative imaginary systems, whereas the mathematical models obtained for these systems using system identification methods need not be negative imaginary. The paper by Mabrok, Lanzon, Kallapur, and Petersen (2013) provides two methods for enforcing negative imaginary dynamics on such mathematical models, given that it is known that the underlying dynamics ought to belong to this system class.

In Bianconi et al. (2013), biochemical transduction networks are modelled through a set of differential equations, and experimentally analysed by measuring a few signals at the end of the cascades. The key idea given in the paper is to introduce a robustness index, the proliferation index and to study its behaviour over the parameter space. The connection between the proposed robustness index and the pathology is analysed.