Overview
This page contains a brief description of mathematical systems and control theory, sometimes shortened to simply “control theory”. This is the organisers’ attempt at a short summary, and we welcome your thoughts. This page will undoubtedly grow and develop over time.
On the one hand, control theory it is the mathematical language of feedback — the interconnection of dynamical systems and their signals. On the other hand, control theory is a framework for describing how systems interact with their wider environment in terms of external input (or control or disturbance) and output (or measurement or observation) signals.
Since in reality virtually no systems are truly closed, control theory is widely applicable. Indeed, control systems and feedback loops are ubiquitous in science and engineering, and arise in traditional areas from aerospace control, manufacturing and process control, through to robotics, electrical power systems, systems biology, and therapeutics. Their study and subsequent use have been fantastically successful, with 31 real-world success stories described in [1]. So important and ubiquitous are control systems that the discipline has been termed a hidden technology [2]. Novel and emerging applications where control systems play an essential role range from the control of autonomous vehicles, smart grids and devices, through to epidemiology [3].
We note that control theory is the mathematical twin of its engineering counterpart control engineering. Both of these fields trace their roots back to the industrial revolution and the requirement to robustly regulate electrical and mechanical systems. Depending on one’s perspective, these two fields are closely related or quite separate.
Given the broad aims of control theory, and the wide-range of areas in which it arises, it is consequently a large subject and celebrated historical results from the field include:
- the Kalman filter [4] for filtering, estimation and prediction, and its contribution to the fields of data-assimilation and signal-processing;
- optimal control theory for posing and solving constrained infinite-dimensional mathematical programs, including the Pontryagin Principle and the Hamilton-Jacobi-Bellman equation (of many possible texts, see, for example [5]);
- the stability radius [6], and its application to distance to matrix instability problems;
- optimal Hankel norm approximations [7] and its connection to so-called Adamjan-Arov-Krein (AAK) theory of Hankel operators.
Both a strength and weakness of control theory is its breadth in that it makes connections to numerous other mathematical areas, including:
- analysis and operator theory via the control of infinite-dimensional systems — such as delay- or partial differential- equations from either a PDE or abstract functional analytic perspective [8];
- calculus of variations and optimisation via optimal control theory and, increasingly prevalently, stochastic optimal control theory [9];
- numerical analysis and numerical linear algebra via singular value- or Krylov subspace- methods for model order reduction [10].
Connections to biology
The above description has presented control theory from a somewhat engineering perspective. However, we quote:
Feedback is a central feature of life. The process of feedback governs how we grow, respond to stress and challenge, and regulate factors such as body temperature, blood pressure, and cholesterol level. The mechanisms operate at every level, from the interaction of proteins in cells to the interaction of organisms in complex ecologies.
M. B. Hoagland and B. Dodson. The Way Life Works, Times Books, 1995.
Indeed, feedback was arguably a biological principle before it was an engineering one. Quite strikingly, these connections were already noted in the mid 19th century by the famous naturalist Alfred Russel Wallace, who likened the theory of evolution by natural selection to that of a feedback system. To quote:
The action of [natural selection] is exactly like that of the centrifugal governor of the steam engine, which checks and corrects any irregularities almost before they become evident; and in like manner no unbalanced deficiency in the animal kingdom can ever reach any conspicuous magnitude, because it would make itself felt at the very first step, by rendering existence difficult and extinction almost sure soon to follow.
On the Tendency of Varieties to Depart Indefinitely From the Original Type, A. Wallace, 1858
Nowadays there are a number of connections between control theory and mathematical biology/ecology, highlighting [11] as an example.
Connections to machine learning
An emerging and exciting application of control theory is to the field of machine learning. We highlight [12] as a recent reference.
References
- T. Samad & A. E. Annaswamy (eds.). The Impact of Control Technology, 2nd edition, IEEE Control Systems Society, 2014.
- K. J. Åström. Automatic Control — The Hidden Technology. In: Frank, P.M. (eds) Advances in Control, Springer, 1999.
- C. Beck et al. Special Section on Mathematical Modeling, Analysis, and Control of Epidemics, SIAM J. Control Optim., 2022, 60(2): Si-Sii
- R. E. Kalman. A New Approach to Linear Filtering and Prediction Problems, J. Basic Eng., 1960, 82(1): 35-45
- A. Seierstad & K. Sydsaeter. Optimal Control Theory with Applications, North Holland, 1987.
- D. Hinrichsen & AJ Pritchard. Stability radius for structured perturbations and the algebraic Riccati equation, Systems Control Lett., 1986, 8(2): 105-113.
- K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error bounds, Int. J Control, 1984, 39(6): 1115-1193.
- M. Tucsnak & G. Weiss. Observation and control for operator semigroups, Birkhäuser, 2009.
- D. Liberzon. Calculus of variations and optimal control theory: a concise introduction, Princeton University Press, 2012.
- A. C. Antoulas. Approximation of large-scale dynamical systems, SIAM, 2005.
- Cowan, Noah J., et al. “Feedback control as a framework for understanding tradeoffs in biology.” American Zoologist 54.2 (2014): 223-237.
- Bensoussan, Alain, et al. “Machine learning and control theory.” Handbook of Numerical Analysis. Vol. 23. Elsevier, 2022. 531-558.