Figure 1. At University of
California, Berkeley, 1978.
I was born in Sydney, Australia but spent the first eleven years on a sheep station in outback New South Wales where my father was the manager. My early education was via correspondence, which at that time meant lesson delivery and response by hardcopy mail. There were days spent in the nearby town of Carinda for socialisation, the school being a single room for all grades. Otherwise, I was subject to my mother’s discipline to make sure the lessons were completed despite the distractions on a 35,000 acre farm. The only sign of any academic talent was that I could finish my lessons very quickly and my mother had no excuse to keep me at a table any longer. After the family moved to Brisbane my main focus for the rest of primary school, according to normal Australian male peer pressure, was catching up to the other kids in playing sports, the latter aided by my having natural sprinting speed. Success here also helped improve my acceptance socially from being the ‘country boy’. Nevertheless, I was easily in the row of top students in my class. (In those days students sat in rows from the least performing at the front near the teacher to the highest at the back.)
Once in a state high school and having mastered some sport credibility, mainly in rugby league football and athletics, I recall a moment where I came third or fourth in the science exams and thought maybe I should study these subjects more seriously. I had the same excellent teacher for mathematics and chemistry, less excellent for physics. I became more interested in mathematics and joined a mathematics group where the teacher set challenges for interest and preparation for competitions. Towards the end of high school I had a dilemma. I enjoyed science, but imagined a career of science being cooped up indoors in labs and offices and here the ‘country boy’ was still there. Images of engineers moving around outside still using maths to build things won out and I chose a University of Queensland degree which started as a combination of electrical and mechanical engineering leaning to the latter.
As the classes progressed my interest changed towards electrical subjects. I simply preferred the maths in circuit theory more than in what was used in the mechanics units. In my third year, I got to know my tutor who I discovered was doing something called a PhD. This really did look exciting and I expressed this at home much to the initial alarm of my father who saw his son doing well enough at university to get a good secure job afterwards. In fact I was the top student by then in electrical engineering headed for a university medal. The subject that excited me most was systems and control which needed differential equations, Laplace transform analysis, linear algebra and optimization just to start. At this point I realised my engineering maths courses had ended up more about recipes than deeper understanding. It was the early 1970’s and EE departments had systems and control research groups with substantial theoretical work being published in mathematics journals and so while motivated by engineering problems the work was essentially applied mathematics. The premier theoretical journals were the Institute of Electrical and Electronics Engineers (IEEE) Transactions on Automatic Control and SIAM Journal of Control and Optimization (JCO) so I joined both IEEE and SIAM to read them.
The need for more maths education was confirmed when I met a potential supervisor for a PhD at the University of Newcastle, Australia where a young Professor Brian Anderson, educated in mathematics and EE plus a PhD in circuit theory at Stanford left his Stanford position to start the new EE department. At that time, he was the youngest professor ever appointed in Australia at 25 years and his career since then in Newcastle and Australian National University has been highly distinguished. This group really was an outpost of top schools in the USA and the UK where researchers with strong theory capability was a feature. I had a personal pressure not to go to the USA for the PhD and this group gave me the opportunity to have the highest standard close to home. In order to prepare, I entered the third year of an honours maths degree, in a class of about 20 students aiming for careers in mathematics. I needed to catch up in fundamentals so had already audited a unit on second year metric spaces which gave me more basics, e.g. epsilon-delta proofs, and read topics like linear algebra now in terms of vector spaces. This was above the level of the standard maths degree which was above the level of the engineering maths I had done so far. The year included units on topology, functional analysis, statistics, differential equations, special functions which all looked useful plus some more fundamental philosophy of mathematics while reading more of what I had missed in the earlier years going back to basic number theory. This was a lot of work but exhilarating.
While completing the Ph.D. degree at the University of Newcastle, Australia, I had a somewhat unusual but positive supervision experience. I started while Brian Anderson was on study leave at UMass, Boston (applying category theory to control system design) and one of his recent PhD students, Peter Moylan, now Lecturer (same as Assistant Professor in the USA), was to help me get started. At that time there were several very interesting threads of system theory to consider, which roughly divided around the use of state-space (first-order differential equations) and input–output (operators on function spaces) methods. Lyapunov stability theory and operator theory gave frameworks in which linear algebra and transfer functions respectively handled the linear time-invariant systems case. The divide was rather clear except for the beautiful Kalman–Yakubovich–Popov Lemma, later called the positive real lemma by Anderson (who gave a multivariable version). This so-called lemma characterized positive-real transfer functions by linear algebraic properties of a state-space representation that, in hindsight, could be seen in terms of the existence of energy- like functions corresponding to the property of passivity. A large stability theory for the classic absolute stability problem for feedback systems grew around these ideas, either operator theoretic in work led by Sandberg, Zames, and Willems or Lyapunov based (including work by Jacques Willems—brother of Jan—with Brockett at MIT and by others). The stability conditions with names like passivity theorem, small-gain theorem, circle criterion, and Popov criterion had some essential similarities in both frameworks. The positive-real lemma played a key role in translating the input-output stability conditions into Lyapunov conditions for the state-space results. However, the absolute stability problem was restricted to a linear plant with nonlinear memoryless feedback usually confined to a sector condition. These results could all be seen as related to restricting the phase and/or gain of the linear plant in generalizations of the Nyquist criterion from fixed gain to nonlinear memoryless feedback.
The thesis aimed to study nonlinear dynamics in feedback and more general interconnected structures following recent generalisations in feedback stability, related optimal control and passivity theory to a large class of nonlinear systems by Anderson and Moylan. As a starting point to the thesis, the task was to obtain a version of the bounded real lemma related to contractive operators and scattering matrices in circuit theory for this class of nonlinear systems. Included in the background reading were the famous two-part dissipativity articles by Jan Willems written while visiting Cambridge University from MIT. (He later established a renowned mathematics group on systems and control in Groningen, The Netherlands.) Reading these papers led to studying his earlier works on input-output stability and the generation of Lyapunov functions from input-output conditions published in SIAM. After my first two papers the work with Moylan proceeded more as a close collaboration than supervision, which became a very good start to my career as an independent researcher. I continued in that theoretical topic for postdoctoral studies at the University of California, Berkeley. The outcomes were on general dissipativity, which included passivity and bounded gain properties in a general quadratic form, giving what a positive-real-type lemma would look like for general dynamical systems, instability results, nonquadratic supply rates, and basic connections (for example, the anticipated result that input-output-stable systems would be globally asymptotically stable with some minor state-space conditions following the well-known linear result was clarified). The papers appeared in the IEEE Transactions, SIAM JCO, the Journal of Franklin Institute and Automatica.
I had arrived in Berkeley on a fellowship that allowed me to keep publishing theoretical results in systems theory. But the group had just won a grant in the Systems Engineering for Power Program set up by the Department of Energy following the energy crisis of the 1970’s. The project was led by Professors Felix Wu and Pravin Varaiya, again strong systems type theoreticians. This program realised challenges in this area needed new ideas beyond traditional power engineering and that naturally came from systems and control groups at MIT, University of Illinois Urbana-Champaign, Berkeley etc. This was an opportunity for me to be included as a postdoc. At this time, I was also thinking about what my career opportunities were, i.e. in the USA/Canada or back in Australia, in engineering or applied maths? The opportunity appeared for me to return to Newcastle after I completed my time in Berkeley by winning a prestigious Queen Elizabeth Fellowship, which I thought I would have a good chance to win, and indeed I was successful. The two lines of thought favoured me looking for a new topic which was more applied oriented and so I joined the power engineering project.
An unsolved problem in electrical power systems was to derive a rigorous Lyapunov function for the so-called classical multi-machine model, which after a network reduction has a second-order model with mass-damper-spring type terms in a mechanics analogy. Incorrect proofs had appeared along with one negative result that the so-called Lure-Postnikov form everyone was looking for (kinetic energy plus potential energy) will not work. The spring term has a non-symmetric Jacobian, which was related to transfer conductances in network terms and so the usual potential term is not a well-defined integral. And attempts to use my stability theory for interconnected systems gave an overly conservative answer. A collaboration was started with Art Bergen, who had asked me to check parts of his new textbook on power systems. He had the benefit of a systems theory background plus intuition derived from industry experience and teaching power systems for many years. This led to a change of outlook. Instead of trying to solve what really was a difficult problem, he led me to reassess the assumptions. We saw that the conductances were a fictitious outcome of the Kron network reduction. Then using a ‘structure preserving model’ with some dynamics on the nodes which were normally eliminated in the classical model, we got a new model where the physical network graph was preserved. Then it was relatively straightforward to derive a rigorous Lyapunov function and proceed to apply Lyapunov stability theory to make a practical stability assessment tool. This then led to a series of papers by us and others with various extensions along with mathematical diversions like developing Lyapunov theory for differential-algebraic systems. As it happens later developments on power grids favoured use of our type of model as the previously eliminated nodes began to host important demand-side dynamics including inverter-based resources such as from solar energy. Incidentally this network view has inspired a series of papers in science journals by physicists on a network science view pf power networks quite different to what would have arisen in engineering in that the network graph becomes the emphasis in studying dynamic behaviour. Interestingly I later saw the original problem compared to a mechanics problem of second-order systems with follower or non-conservative forces which indeed presented theoretical challenges in that area. The lesson here was that while it would have been mathematically heroic to continue to try and solve the original problem, it was easier to solve a revised and considerably more useful problem - and I mean theoretically more useful as well as practically. As far as I know the original and now obsolete problem for the classical model remains to be solved – if indeed there is a solution in any useful sense considered so far - some helpful contributions notwithstanding. The other thing I learned here was that despite having a theoretical toolbox I should not be the ‘hammer looking for a nail’ when confronting a new problem. My previous work in stability theory was no doubt helpful in spirit, but not in providing the actual solution to this new problem in power systems.
Figure 2. Slide from power system stability lecture.
Back in Australia and in the aforementioned group in systems and control in Newcastle, I had to teach something. Given control was in other hands I was happy with electrical circuits and power systems, the latter at an advanced level and so could be connected to my research. With a personal connection to Sweden, I ended up spending a year at the famous control group in Lund just after a major system collapse. The engineers had a good physical understanding of what happened, but no methods of analysis for what was a relatively new phenomenon with voltages collapsing instead of the usual mechanisms of instability. My Berkeley experience had taught me not to immediately start analysing whatever model was already in place. After talking to engineers I wrote my own model and once again the analysis was initially not hard, but gave immediate insights. The work did lead to more mathematically demanding analysis as we went deeper in terms of considering different bifurcations as a way to analyse the collapse phenomena. The initial analysis was already useful in setting up protection schemes to prevent such collapses and this taste of using science to achieve a good industry outcome was becoming appealing for my future work.
Henceforth as a professor leading a group, I had diverse activity from theoretical papers in SIAM to engineering consulting in industry. I saw systems and control is a rigorous point of view that may or may not end up in theorems depending on the problem at hand. Some of the more practical work has led to work in stability theory in a program I called ‘progress to complexity’ where switching, impulsive events etc motivated by practicalities are tackled theoretically. Here again once you get the models looking right it is inevitable that the initial work (or low hanging fruit) will be useful and easier. Special situations will be more mathematically challenging, hopefully still useful at some time, but maybe not by me or ever.
Nevertheless, there is always this tension between what is mathematically challenging and what is worthwhile for applications. While I achieved some overlap in the power systems area, working with two modes has led to me being elected to engineering (including IEEE) and science (including SIAM) elected fellowships and other awards based on power systems engineering and systems and control theory. I like the quote that captures what a theoretically inclined engineering senior colleague told me (but later I learned actually comes from psychology) that there is ‘nothing as practical as a good theory’ and I might add ‘with the right model’. On visits to Russia I learned about their intense debates about different approaches of starting with the practical vs the abstract to get good mathematics. Incidentally the well-known strength but isolation of Soviet mathematics was seen first hand in my work after visits there in finding that a paper in an obscure St Petersburg journal had achieved an energy integral and some preliminary Lyapunov analysis without Kron reduction years before my efforts, very different but certainly noteworthy as an earlier result.
I feel privileged to have had the opportunity to work in both applied maths and engineering which I see as overlapping. This is certainly true in the systems and control area which I chose for my career. Whether mathematics or engineering researchers dominate a research topic can be less about maths rigour than the history of the subject and the people concerned. In retirement I am still doing research now on a new era of stability and control problems related to the clean energy transition. And for fun at the desk I am drawn back to playing with basic mathematics questions probably similar to those I studied in that high school group, always with a sense that mathematics is beautiful as well as useful.