Nicos Karcanias – an obituary

1. Introduction

Nicos Karcanias was a leading figure in the control systems field for more than 40 years. His contributions in the area of linear multivariable systems and control systems design have been text-book material since the early 1980s. His research investigations have strong thematic unity in both technique and problem formulation as well as a broad context. A central feature in his work was the exploitation of system structure, as characterised particularly by discrete invariants, for both analysis and design. One of his major interests was the application of systems theory to provide an integrated design philosophy for evaluation, synthesis and instrumentation, particularly in the context of process control. Nicos was also interested in the contributions of Ancient Greece to mathematics, science and technology, and co-authored a number of papers about mythological and real automata described by Homer and other ancient authors, and about early forms of mathematical modelling. Nicos was a much-loved family man and, together with his beloved wife Themis, cultivated a wide circle of warm friendships. He had deep interests in politics and was an unashamed gastronome, keenly preparing as well as consuming dishes and preserves.

2. Personal life

Nicos's birthplace was Messolonghi, Greece – a small and quiet provincial town with a famous and glorious historical past. He was born on 21 April 1948, the eldest of three siblings. In the neighbourhood where he grew up, scientists and ordinary fishermen lived close to one another – just below his home was the home of the former prime minister of Greece, Charilaos Trikoupis, and further along was the house of the poet Kostis Palamas – these people were inspirational to Nicos and his peers.

Nicos was a lively child, with a love for life and many interests. He enjoyed water games, fishing, boating in the famous lagoon of Messolonghi, and endless hours of football in the ‘saltsina’ (areas where the sea brush had dried up) next to the lagoon. He was also a very diligent student, insatiable in reading in history, geography and mathematics, his father being the headmaster and his aunt a teacher. He was a diligent church-goer and assistant to the priest as a child of the Sanctuary. His beloved cats Daniel and Tatabu were the most vulnerable cats in the neighbourhood.

In 1962, Nicos's father was appointed to a new job as headmaster of a school in Athens, and the whole family moved to the very central district of Theseion, close to the Acropolis, where they lived in an old neo-classical house. In his studies, Nicos began to distinguish himself particularly in mathematics, physics, chemistry. Even at this young age, Nicos was never satisfied with a superficial approach to problem solving. Nicos and his close friend Kostas Giannopoulos were the fear of their Mathematics teacher. Sometimes indirectly and sometimes directly, he asked for their opinion on the solution of a problem that was beyond his knowledge. From 1964, Nicos began to prepare for the very challenging entrance exams at the ‘Polytechnion’ (the National Technical University of Athens) by attending the Heraklitos private school in addition to his normal schooling. In September 1966, he won a scholarship, having been placed first in the entrance examination, to study Mechanical and Electrical Engineering at the Polytechnion.

A very traumatic event in Nicos's life was the seizure of power by a repressive military junta on 21 April 1967 – Nicos's 19th birthday, as his friends would often point out to him in later years. Many professors at the Polytechion resigned or were made redundant due to their political views. In January 1969, a new professor in the field of Circuits and Systems, Ioannis Diamesis, was elected, who inspired students to further their studies in Automation and Control – Nicos was one of them. The weekly programme at the Polytechnion was very demanding, typically 48–55 hours of lectures, laboratory exercises and supervisions. As his close friend Tryfon Kousiouris recalls: among his fellow students there is the memory of a charming young man, and in a class that resembled a battalion of Jesuit monks, Nicos was usually met by a girl after school!

In the summer of 1970, as part of an exchange of students through IAESTE, Nicos and Tryfon were offered work experience in France. Unfortunately Nicos's work at a corn company in Haubourdin, a suburb of Lille, had little to do with engineering and this was very disappointing for him. But this was compensated for by staying in the house of a very kind lady, a member of the Socialist Party, and there he made a number of life-long friends. Back in Greece in 1971–72, he gained valuable industrial experience in the design of servomechanisms as Research Engineer in the Nuclear Research Centre DEMOCRITUS, Athens.

After graduating from the Polytechneion in 1972, he left Greece for England, where he got a Master's degree in Control Engineering from UMIST (University of Manchester Institute of Science and Technology) in 1973. He impressed his fellow students not only with his academic abilities but also as a kind and gentle person who could be very passionate, not only about his studies but also about the political and social issues of the day. He continued on at Manchester for the PhD, under the supervision of Professor A.G.J. MacFarlane. It was Christmas 1973, during a short visit to Athens, that Nicos met his future wife Themis at a friend's house – his bright personality, sense of humour, and Che Guevara style with beret and strong political views captured her attention! Themis joined Nicos in Manchester in the summer of 1974. As they went around the old industrial sites of Manchester and saw first hand the remnants of the industrial revolution, a new world opened up for them. They soon became a close and inseparable couple.

Nicos and Themis moved to Cambridge in the summer of 1974 after Alistair MacFarlane's appointment to the chair in control at Cambridge. Cambridge was a city of their dreams. Nicos, always dedicated to his academic work and an early bird, would cycle every morning to the Engineering Department to work on his PhD thesis, though he never neglected his other activities and many interests. As young people, their life in Cambridge was a most exciting period. They would go from interesting lectures, to political demonstrations, to endless political discussions, to musical festivals and of course to parties! Their lifestyle reflected their international outlook on life. Nicos and Themis built together a harmonious and happy family, always surrounded by people.

While still in Manchester, Nicos was active in political opposition to the Greek junta and developed a strong interest and ideological conviction in socialism – he would call it ‘Democratic Socialism’ to distinguish it from the other kind. In Cambridge, he could often be seen selling the ‘Socialist Worker’ in the market square on Saturday mornings. After the award of his PhD from the University of Manchester in June 1976, Nicos was called up by the Greek army (the junta having fallen from power in 1974). He served for 3 months and returned to Cambridge to take up a post-doctoral research assistant post with Alistair MacFarlane.

In 1980, Nicos became Lecturer in the Department of Systems Science at the City University in London. He remained at City University for the rest of his career, although he continued to live in Cambridge. He subsequently moved to the Electrical Engineering Department where he was promoted first to Reader in Control Theory in 1986 and then to a personal Chair in 1993 as Professor of Control Theory and Design. He established the Control Research Centre in 1990 and later the Systems and Control Research Centre in 2000; he remained its Director for the rest of his career. From 2003, he was Associate Dean for Research in the School of Engineering and Mathematical Sciences.

Countless staff and students of City remember Nicos as one of the first people who welcomed them to the University. Their first impression was of a gentleman – generous, always cheerful and who took an interest in them. They remember their pleasant chats in the corridor or entrance hall. Nicos committed his life to serving City and guided the intellectual development of numerous masters and doctoral students in the University. He was notable in his support of both senior and junior colleagues. He engaged tirelessly with many administrative tasks such as preparing several Research Excellence Framework submissions for the department and working closely to make the best case for engineering which was rewarded with improving scores for the department over the years.

Nicos's office door was always open, inviting colleagues and students to drop in. A Greek coffee, or tea, and in the early years a cigarette, were on offer, and perhaps even, on special occasions, a drop of Ouzo (like nectar as he would say).

3. Research

3.1. Geometric theory of zeros

Nicos arrived in Manchester at an exciting time in the development of the control systems field. Barely a decade previously the state space approach to control had emerged with its strong foundation in linear algebra and new concepts such as controllability and observability. The linear quadratic regulator and Kalman filter had established themselves as ‘modern control’ design methods and achieved spectacular success in the Apollo programme and the moon landings. At this time, the Control Systems Centre at UMIST was a leading international centre for control research – Rosenbrock's system theory was a link between the state space and frequency response approaches, and Rosenbrock and MacFarlane were developing multivariable generalisations of the classical frequency response design methods.

Nicos's PhD was at the centre of emerging ideas at this time. His thesis, entitled ‘Geometric theory of zeros and its use in feedback analysis’ (1975), was a bridge between the algebraic methods of Kalman's approach and frequency response thinking as exemplified in the search for a better understanding of zeros in multivariable systems (Karcanias, 1975; MacFarlane & Karcanias, 1976). In Rosenbrock's system theory (Rosenbrock, 1970), a central object is the ‘system matrix’ which takes the form (1) $P(s)=\left[\begin{array}{cc}sI-A& -B\\ C& D\end{array}\right]$(1) for the standard (continuous-time) linear state space system (2) $\left[\begin{array}{c}\dot{x}\\ y\end{array}\right]=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{c}x\\ u\end{array}\right],$(2) where s represents either the Laplace transform variable or the differential operator $\mathrm{d}/\mathrm{d}t$. In (2(2) $\left[\begin{array}{c}\dot{x}\\ y\end{array}\right]=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{c}x\\ u\end{array}\right],$(2) ), $x(t)$, $u(t)$ and $y(t)$ are the state, input and output of the dynamical system, and A, B, C and D are real matrices of compatible dimension. Rosenbrock introduced the notion of transmission and decoupling zeros defined in terms of loss of rank in the system matrix or sub-matrices. Nicos introduced the notion of invariant zeros which are defined as the set of zeros of the invariant polynomials in the Smith form of $P(s)$. Nicos was influenced by the geometrical concepts introduced by Wonham and Morse (1970) and explored the physical meaning of system zeros, in particular the idea of characteristic directions as well as frequency in analogous way to system poles. The main emphasis was on the physical interpretation of invariant zeros in terms of the general zero-output behaviour of a linear dynamical system. For the simple case of D = 0, Nicos showed that the ‘zero directions’ in the state space corresponding to the invariant zeros belonged to $(A,B)$-invariant subspaces which are in the kernel of the C matrix.

Nicos's work on zeros played an important role in linking the frequency response and state space approaches to control. Beyond that it paved the way to understand performance limitations in controller design based on system invariants, which was a recurring theme throughout his research career.

3.2. The squaring down problem

A problem introduced in Nicos's PhD is the ‘squaring down’ problem: given a system matrix (1(1) $P(s)=\left[\begin{array}{cc}sI-A& -B\\ C& D\end{array}\right]$(1) ) in which D = 0 and CB is a ‘tall’ matrix, i.e. there are strictly more outputs than inputs, find a matrix L so that $LC=C_1$ where $C_1$ is a square matrix so that the system matrix (3) $P(s)=\left[\begin{array}{cc}sI-A& -B\\ C_1& 0\end{array}\right]$(3) has invariant zeros at prescribed locations. In case the zeros cannot be placed at arbitary locations determine what locations are ‘assignable’. Nicos approached the problem geometrically, to assign a set of frequencies $s_i$ and corresponding vectors $x_i$ to form $(A,B)$-invariant subspaces in the kernel of LC, and to derive conditions in geometric form for an assignment.

Nicos's introduction and approach to the squaring down problem was important for his research vision for two reasons. First, the problem was a prototype for a more general type of determinantal assignment problem that he introduced and studied later with students and co-workers. Second, it can be seen as an example of his philosophy of system design whereby an early step in the process is to arrive at a plant which is as easy as possible to control.

3.3. Matrix pencils

A mathematical tool which was central in Nicos's work was the theory of matrix pencils; a matrix pencil takes the form $s{H}_1+H_2$, where s is a variable but $H_1$ and $H_2$ are constant matrices. In an early paper together with Kouvaritakis, Nicos developed further the basic notion of a zero in terms of ‘output zeroing’ to relate internal and external behaviour of a system (Karcanias & Kouvaritakis, 1979). The zero pencil sNM−NAM was introduced, where N is a full rank left annihilator of B and M is a basis matrix representation of Ker(C), to develop the concept of geometric dimension of zeros, exploiting the algebraic (Kronecker) structure of the zero pencil, and its associated eigenstructure.

Nicos extended these ideas together with Kalogeropoulos to generalised linear systems classifying algebraic structure, regularisability and normalisability properties under generalised state feedback, with new feedback invariants identified in terms of a restriction pencil (Karcanias & Kalogeropoulos, 1989). New sets of invariants were introduced with Leventides in terms of the Grassmann representatives of the column and row spaces of the pencil and the associated Plücker matrices and applied to the study of singular systems (Karcanias & Leventides,1996).

The theory of matrix pencils was an inspiration for Nicos as it was a unique vehicle for many of his ideas: notions of equivalence; linking of algebraic and geometric concepts; the theory of invariants. Aside from the contributions to control that arose from it, the theory served to bring to a wider audience the historical literature in linear algebra and algebraic geometry with a closer connection to modern computational methods.

3.4. Zeros at infinity and almost $(A,B)$-invariant subspaces

Matrix pencils played a key role in Nicos's approach to a research problem that was much studied in the field in the early part of his career. This concerned the behaviour a system exhibits ‘at infinity’, namely when $|s|$ becomes arbitrarily large. One motivation for this was that root-locus methods were an effective design tool for single-input/single-output (SISO) systems, and their asymptotic behaviour was governed by the system structure at infinity – which in SISO systems is no more than the difference between the numerator and the denominator degrees of the transfer function, but in the multivariable case is more complex. In studying this problem together with S. Jaffe, Nicos developed the notion of ‘almost $(A,B)$-invariant subspaces’ of Willems and established the equivalence between the algebraic, matrix pencil, characterisation of the sub-spaces of an ‘extended’ geometric theory and their dynamic characterisation. A complete classification of almost $(A,B)$-invariant, $(A,B)$-invariant, almost controllability and controllability subspaces was derived in terms of matrix pencil invariants (Jaffe & Karcanias, 1981). The approach explained the importance of the infinite elementary divisors as associated with ‘asymptotic’ or infinite spectrum subspaces and their crucial role in the asymptotic behaviour of the root-locus.

3.5. Determinantal assignment problem

A standard problem of state space theory was ‘pole assignment by state feedback’. Suppose one applies state feedback of the form u = Kx, where K is a constant matrix, to the system (2(2) $\left[\begin{array}{c}\dot{x}\\ y\end{array}\right]=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{c}x\\ u\end{array}\right],$(2) ), then how may the roots of $det(sI-A-BK)$ be placed in desired locations by choice of K? Solutions to this problem were worked out fairly quickly. But what if only the output is available for feedback? Assigning the roots of $det(sI-A-BKC)$ is much more difficult. An alternative way to write this problem is to use the matrix-fraction form of the transfer function $N_r(s)D_r(s{)}^{-1}$ (or $D_{\ell }(s{)}^{-1}{N}_{\ell }(s)$), where $N_r(s)$, $D_r(s),$ etc. are polynomial matrices. The closed-loop poles are then given by the roots of (4) $\begin{array}{rl}& det[D_r(s)+K{N}_r(s)]\\ & =det\left\{\left[\begin{array}{cc}I& K\end{array}\right]\left[\begin{array}{c}{D}_r(s)\\ N_r(s)\end{array}\right]\right\}\triangleq det[HT(s)].\end{array}$(4) For a given $m\times p$ polynomial matrix $T(s)$ ($m\ge p$), the problem of finding a constant $p\times m$ matrix H such that $det[HT(s)]=a(s)$ for some given Hurwitz polynomial $a(s)$ was called by Nicos the Determinantal Assignment Problem (DAP). The DAP unifies various frequency assignment problems of linear control theory, including output feedback pole placement, ‘zero assignment’ and the squaring down problem. Nicos's approach to the DAP makes use first of the Binet–Cauchy theorem to rewrite the defining equation of the DAP as (5) ${\mathbf{h}}^{T}P={\mathbf{a}}^T,$(5) where $\mathbf{h}$ is the exterior product of the rows of H (equivalently the vector of highest order minors of H arranged lexicographically), P is a constant matrix derived from the exterior product of the columns of $T(s)$ and $\mathbf{a}$ is the vector of coefficients of $a(s)$. The problem is to determine if there is an ${\mathbf{h}}_0$ in the linear space of solutions $\mathbf{h}$ of (5(5) ${\mathbf{h}}^{T}P={\mathbf{a}}^T,$(5) ) such that the exterior equation ${\underset{\_}{h}}_1\wedge {\underset{\_}{h}}_2\wedge \dots \wedge {\underset{\_}{h}}_p={\mathbf{h}}_0$ can be solved. The solvability of this exterior equation is referred to as decomposability of ${\mathbf{h}}_0$, and it is characterised by the set of quadratic Plücker relations describing the Grassmann variety of a corresponding projective space. Work with students including Christos Giannakopoulos and John Leventides led to new tests and methods for decomposability, necessary and sufficient conditions in terms of the rank properties of an object they introduced, the Grassmann matrix, as well as computational methods (Giannakopoulos & Karcanias, 1985; Karcanias & Giannakopoulos, 1984, 1988; Karcanias & Leventides, 2016). Nicos's programme of work on this problem was exemplary in bringing to the foreground some techniques and literature from exterior algebra and algebraic geometry that Nicos had promulgated from the time of his PhD in Manchester.

3.6. Complex systems and structure evolving systems

In the last 20 years or so of his career Nicos developed a growing interest in the modelling of (linear) systems viewed more broadly than the design of a controller for a given system but rather the design and evolution of the system as a whole (Karcanias, 2008). Typically there are many detailed decisions that are made when assembling complex systems, such as selection and location of sensors and actuators, and more generally selection of components, design of subsystems and their interconnection, which determine the final form of the model. Thus, rather than taking (1(1) $P(s)=\left[\begin{array}{cc}sI-A& -B\\ C& D\end{array}\right]$(1) ) as a starting point, Nicos became concerned with the details of how to make such decisions, so that the final system has desirable dynamic properties – being easy to control, for example.

Nicos's approach was to develop a framework that was broad enough to encompass all of the design choices throughout the life-cycle of the design. An essential idea is to assess performance limitations as a function of choices in dynamic elements and interconnection topology. In this way, his approach connected closely with his other system-theoretic results while his European projects on topics such as ‘Systems of systems’ and ‘Advanced manufacturing’ provided the collaboration and applications opportunities for the programme.

3.7. Technology of Ancient Greece

As Nicos pointed out ‘automaton’ ($\alpha \upsilon \tau \acute{o}\mu \alpha \tau o$) is a Homeric word that first appeared in the Iliad (Vasileiadou et al., 2003). There one finds early visions of concepts and technology: the automatic gates of heaven, automatic tripods and adaptive bellows; and in the Odyssey there are the automatic ships of the Phaeacians and Egyptians. Likewise the Greek word $\sigma \acute{\upsilon }\sigma \tau \eta \mu \alpha$ ‘system’ comes from the ancient verb $\sigma \upsilon \nu \acute{\iota }\sigma \tau \eta \mu \iota$ which occurs frequently in ancient Greek written sources in many contexts with a basic meaning that the whole is made from parts and that the whole may have properties not always attributed to the individual parts. With Soultana Vasileiadou, Nicos highlighted what seems to be the first recorded definition of a system by the Pythagorean Kallicratides – any system consists of contrary and dissimilar elements, which unite under one optimum and return to the common purpose – and traced the idea through Hippocrates' holistic approach to medicine and Aristotle's system of logic (Karcanias & Vasileiadou, 2007).

Nicos was fascinated with the science, technology and philosophy of Ancient Greece, and in the precursors that one can find of modern ideas of systems and control. This meshed perfectly with his view of his discipline as extending far beyond engineering into the social sciences, biology, economics, politics and the environment. For Nicos, understanding the historical roots of his discipline demonstrated that the theory and practice of systems and control required ‘not the specialisation, but the combination of all the branches of sciences, as well as their connection with philosophy and arts’ (Karcanias & Vasileiadou, 2007). It is in such a broad canvas that his intellectual life and work are best summed up.

4. Epilogue

Nicos was a loyal, kind and supportive friend to many people. He had the impressive ability to be constantly pleasant and likeable and with the best sense of humour. ‘Retirement’ was something that existed only in the abstract for him. Devotion to his field of study and responsibility for his students, co-workers and University were completely independent of his stage of employment. Nicos was always brimming with ideas and optimism for new paths to be explored and discoveries to be made. If there is regret that some of this is cut short there is the certain fact that his ideas, inspiration and example live on strongly among his many colleagues and students around the world.

Nicos died at home in Cambridge on 17 April 2020 after a long battle against cancer. He is survived by his wife Themis, his children Alexandra and Aris, and his grand-daughter Stella.

5. PhD students of Nicos Karcanias

1. Christos Giannakopoulos, Frequency assignment problems of linear multivariable systems: an exterior algebra and algebraic geometry based approach, 1984.

2. Grigoris Kalogeropoulos, Matrix pencils and linear system theory, 1985.

3. Vasilis Laios, A unified approach to decentralised control based on the exterior algebra and algebraic geometry methods, 1990.

4. David R. Wilson, Algebraic synthesis methods for linear multivariable systems: decentralised stabilisation, 1990.

5. Marilena Mitrouli, Numerical issues and computational problems in algebraic control theory, 1991.

6. M. Hadad Zarif, Structural properties of linear systems, 1992.

7. Xu Yi Shan, Bounded feedback and structural issues in linear multivariable systems, 1992.

8. John Leventides, Algebrogeometric and topological methods in control theory, 1993.

9. Helen Eliopoulou, A matrix pencil approach to the study of geometry and feedback invariants of singular systems, 1994.

10. E. Milonidis, Finite settling time stabilisation for linear multivariable time-invariant discrete-time systems: an algebraic approach, 1994.

11. Elisa G. Lampakis, Algebraic synthesis methods for linear multivariable control systems, 1995.

12. Dimitris Vafiadis, Algebraic and geometric methods and problems for implicit linear systems, 1995.

13. N. Tamvaklis, Sampling and structural properties of discretised linear models, 1999.

14. Ermioni Topintzi, System concepts and formal modelling methods for business processes, 2001.

15. Soultana Vasileiadou, Evolution of system, modelling and control concepts in Ancient Greece, 2002.

16. Daniel Nankoo, Linear systems and control structure selection, 2003.

17. Konstantinos G. Vafiadis, Systems and control problems in early systems design, 2003.

18. Stavros Fatouros, Approximate algebraic computations in control theory, 2003.

19. Konstantina Milioti, Conceptual modelling and systems theory with an application using real options analysis, 2007.

20. E. Sagianos, Structural identification and the optimal assignment problem, 2008.

21. Athanasios Papageorgiou, Strong stability of internal systems descriptions, 2009.

22. Giorgos Galanis, Dynamic polynomial combinants and linear systems, 2010.

23. Dimitrios Polyzos, Measuring system properties & structured diagnostics for the selection of sensors, actuators placement & eigenstructure sssignment, 2010.

24. Ivan Lucic, Risk and safety in engineering problems, 2010.

25. Dimitris Christou, ERES methodology and approximate algebraic computations, 2011.

26. Georgios Grigoriou, Structure evolving systems: model structure evolution and system properties, 2012.

27. Athanasios Pantelous, Solutions properties and techniques for implicit systems, 2013.

28. Hatim Ibrahim Elsayed, Utility applications of smart online energy systems: a case for investing in online power electronics, 2014.

29. Haotian Zhang, Smart grid technologies and implementations, 2014.

30. George Petroulakis, The approximate determinantal assignment problem, 2015.

31. Ali A. Sayyad, Optimisation of condition number for eigenstructure problems, 2016.

32. Ioannis Meintanis, Structural and computational aspects of determinantal assignment problems for multivariable systems, 2016.

33. Maria Livada, Implicit network descriptions of RLC networks and the problem of re-engineering, 2017.

34. Afrouz Farshad Mehr, Determination of design of optimal actuator location based on control energy, 2018.

Acknowledgments

The authors are grateful for the comments, remarks and reflections of family, friends and colleagues of Nicos including: Themis Karcanias, Nasos Karcanias, Grigoris Kalogeropoulos, Kostas Giannopoulos, Tryfon Kousiouris, Paul Taylor, Ken Grattan, Maria Tomas-Rodriguez, Ivan Lucic, Christos Giannakopoulos, John Leventides, Dimitris Vafiadis, Soultana Vasiliadou, Nicos Christodoulakis, Martin Newby, Basil Kouvaritakis.