Excerpt

The origins of mathematical control theory can be traced back to James Clerk Maxwell’s 1868 paper “On Governors,” which phrased the question of stability of James Watt’s centrifugal governor in terms of a numerical criterion involving roots of polynomials. The early days of feedback control as a separate discipline saw widespread use of frequency domain methods based on Fourier and Laplace transforms, with pioneering contributions by Hendrik Bode (1940), Harry Nyquist (1932), and others during the 1930s–1940s. The underlying paradigm viewed systems in terms of their external (input/output) behavior, which was described mathematically using so-called transfer functions, that is, proper rational functions of a single complex variable. The criterion of (external) stability, worked out by Maxwell for second- and third-order systems and later in full generality by Edward Routh (1877) and Adolf Hurwitz (1895), pertained to the poles of the transfer function. A system described in this way receives inputs (or stimuli) from its environment, reacts by producing outputs (or responses), and can be controlled by means of output-to-input feedback. This stimulus–response view was rather influential even beyond control theory, for example, in psychology and physiology, and it was adopted by Norbert Wiener (1948) in his book on cybernetics. The extension of these ideas to stochastic systems, where stationary processes were modeled in the frequency domain by their spectral densities, led to the seminal results on linear filtering by Andrey Kolmogorov and Wiener.

However, a number of limitations of this paradigm soon became apparent. The most obvious one was that it dealt primarily with time-invariant systems, that is, systems whose input–output behavior did not depend on time explicitly. Classical transform-based methods were not equipped to handle time-varying systems in a straightforward way, and the mathematical and computational challenges involved in extending the Kolmogorov–Wiener filtering theory to nonstationary processes were rather considerable. In addition, the classical theory was developed for single-input, single-output systems, whereas in many engineering applications (such as chemical process control) one would routinely deal with systems where several output variables had to be regulated simultaneously by manipulating several input variables. The ad hoc approach of reducing these multivariable control problems to a series of single-input, single-output problems lacked principled justification. Finally, stability analysis based on external input–output behavior could not account for the presence of internal instabilities, which in turn could not be compensated using output-to-input feedback. Taken together, all of this was evidence that the field of control systems was facing what Thomas Kuhn referred to as a crisis in his book The Structure of Scientific Revolutions, that is, the situation where the explanatory framework based on the current paradigm starts breaking down as various anomalies accumulate.

Bibliography

Bechhoefer, J. 2021. Control Theory for Physicists. Cambridge, UK: Cambridge University Press.

Bode, H. W. 1940. “Relations between Attenuation and Phase in Feedback Amplifier Design.” Bell Systems Technical Journal 19 (3): 421–454. https://doi.org/10.1002/j.1538-7305.1940.tb00839.x.

Hermann, R., and A. J. Krener. 1977. “Nonlinear Controllability and Observability.” IEEE Transactions on Automatic Control 22:728–740. https://doi.org/10.1109/TAC.1977.1101601.

Hurwitz, A. 1895. “Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt.” Mathematische Annalen 46 (2): 273–284. https://doi.org/10.1007/BF01446812.

Kálmán, R. E. 1960a. “Contributions to the Theory of Optimal Control.” Boletín de la Sociedad Matemática Mexicana 5:102–119.

—. 1960b. “On the General Theory of Control Systems.” IFAC Proceedings Volumes 1 (1): 491–502. https://doi.org/10.1016/S1474-6670(17)70094-8.

Lin, J.-T. 1974. “Structural Controllability.” IEEE Transactions on Automatic Control 19:201–208. https://doi.org/10.1109/TAC.1974.1100557.

Maxwell, J. C. 1868. “On Governors.” Proceedings of the Royal Society of London 16:270–283. https://doi.org/10.1098/rspl.1867.0055.

Nyquist, H. 1932. “Regeneration Theory.” Bell Systems Technical Journal 11 (1): 126–147. https://doi.org/10.1002/j.1538-7305.1932.tb02344.x.

Routh, E. J. 1877. A Treatise on the Stablity of a Given State of Motion, Particularly Steady Motion. London, UK: McMillan.

Shields, R. W., and J. B. Pearson. 1976. “Structural Controllability of Multi-Input Linear Systems.” IEEE Transactions on Automatic Control 21:203–212. https://doi.org/10.1109/TAC.1976.1101198.

Shinbrot, T., C. Grebogi, J. A. Yorke, and E. Ott. 1993. “Using Small Perturbations to Control Chaos.” Nature 363 (6428): 411–417. https://doi.org/10.1038/363411a0.

Sussmann, H. J., and J. C. Willems. 1997. “300 Years of Optimal Control: From the Brachystochrone to the Maximum Principle.” IEEE Control Systems Magazine 17:32–44. https://doi.org/10.1109/37.588098.

Wiener, N. 1948. Cybernetics: Or Control and Communication in the Animal and the Machine. Cambridge, MA: MIT Press.

Willems, J. C. 1972a. “Dissipative Dynamical Systems Part I: General Theory.” Archive for Rational Mechanics and Analysis 45:321–351. https://doi.org/10.1007/BF00276493.

—. 1972b. “Dissipative Dynamical Systems Part II: Linear Systems with Quadratic Supply Rates.” Archive for Rational Mechanics and Analysis 45:352–393. https://doi.org/10.1007/BF00276494.

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