Toward an Algorithmic Control Theory

W HAT does it mean to compute? For the ancient Greeks of the Pythagorean school, a “legitimate” computation required only a ruler and a compass; for the early 9th century Persian mathematician al-Khwarizmi, it was “completion” and “balancing” of equations. In the absence of infinitesimal calculus (a theory that would not become available until the late 17th century), and influenced by the Greek and Persian schools of thought, most problems considered tractable/solvable for centuries involved geometry and algebra. The advent of computers§ and associated algorithms have changed all that, expanding what it means for a problem to be “solvable”. Today, we do not give a second thought on whether answers provided by a computer are legitimate, even when “computing” is embedded in the operation of engineered systems. But should we? As casual users of personal computers and calculators in our daily life, we have become accustomed to adopting the numbers spit out by these devices as the indisputable truth. We embrace them as the answer to our query, without second guessing, although we all know very well that this is a fallacy. We routinely use radicals and trigonometric expressions in our formulas but we tend to forget that we never compute them exactly. Computed values of 2 p or sin 56 deg, for instance, are rational approximation surrogates of these quantities, which we take ipso facto to be their “true values” for all practical purposes— or not. ¶ This leap of faith ofwhat an “answer” to an algebraic equation should be is firmly grounded on the certainty and precision by which this answer is provided: the calculated value is close to its true value with an arbitrary degree of accuracy. In addition, this value can be computed deterministically, after a finite number of (known a priori) floating-point operations. Let us now consider the task of computing a simple control action, namely, the design of a controller to generate such an action via state feedback. Feedback control design, by and large, focuses on finding a mapping φ:X → U that assigns to each plant state x ∈ X a control action u ∈ U so that the controlled plant under the action u φ x has the desired performance. In many cases, the mapping φ is required in “closed form”, often as the result of a laborious offline process. The sole task of the onboard computer is to reliably implement this precomputed mapping. There is a price to pay, however, for restricting the search of state-feedback strategies to such “nice” closed-formmappings; to beginwith, such amappingmay not even exist. The French mathematician Évariste Galois (1811–1832), for instance, in his celebrated 1846 paper “Oeuvres Mathématiques” (published posthumously after finally Liouville reviewed his manuscript and declared it to be correct) showed that no polynomial of degree five or higher has a solution that can be computed in analytic form, as a function of the polynomial coefficients. ** Of course, nowadays no one considers the evaluation of the roots of a polynomial (of any order! ) to be an issue. Computers have thus redefined what problems are considered solvable. By expanding what we may consider an acceptable feedback control strategy, we therefore open the stage for a larger class of feedbackmappings that are, perhaps, the result of an iterative process (i.e., an algorithm). As long as these mappings can be computed with predetermined precision after a finite number of iterations in a timely manner (very much like the calculation of 2 p or sin 56 deg on a handheld calculator), these are as good as closed-form algebraic expressions from X to U. To stress this point further, consider for a moment the case of LQR control design. If linear quadratic theory had not been introduced by R. Kalman in 1950s, but rather long before computers could efficiently solve matrix Riccati equations, it is doubtful whether the impact of the beautiful LQR/LQG theory would have been so profound. In all likelihood, it would have remained exactly that: a beautiful theory. Certainly, the linear quadratic problem would not have been considered solvable (for engineering purposes), at least not more so than a fifth-order polynomial would have been considered solvable in the late 19th century. Fortunately, this is not the case. The simultaneous development of reliable numerical algorithms (and computer technology) that could efficiently solve matrix Riccati equations elevated the theory to one of the most useful control engineering tools ever. Nonetheless, we argue that there is a lot of room for improvement. Over the years, computers and computer algorithms have redefined and extended the meaning of “solvability” in engineering. Let us not forget that computational methods, parallelization of numerical algorithms, and new computer architectures and hardware (GPUs, multicore CPUs, FPGAs) have revolutionized many engineering disciplines in the past 40 years. Many traditional fields, such as fluid mechanics and structures, have embraced the use of computers and numerical methods many years ago. The use of computational fluid dynamics, finite elements, etc., has had an enormous impact on solving real-world problems. By comparison, control theory, by and large, has been somewhat reluctant to incorporate purely numerical methods for deriving (as opposed to just implementing) feedback controllers on the fly. Granted, what distinguishes control from most other disciplines is that, although computations for solving fluid and structure problems can be performed offline, (feedback) control, on the other hand, requires online computation of the respective control actions. Going from solving offline the equations for the controller gains to the online Received 14 January 2017; revision received 15 January 2017; accepted for publication 16 January 2017; published online 6 February 2017. Copyright © 2016 by Panagiotis Tsiotras and Mehran Mesbahi. Published by the American Institute ofAeronautics andAstronautics, Inc., with permission.All requests for copying and permission to reprint should be submitted to CCC at www.copyright. com; employ the ISSN 0731-5090 (print) or 1533-3884 (online) to initiate your request. See also AIAA Rights and Permissions www.aiaa. org/randp. *Professor, Daniel Guggenheim School of Aerospace Engineering; tsiotras@gatech.edu. Fellow AIAA. Professor, William M. Boeing Department of Aeronautics and Astronautics; mesbahi@aa.washington. edu. Senior Member AIAA. And by a fortuitous coincidence, the Guest Editorial duo of this Special Issue reflects both cultural heritage. §And not just digital ones. The Abacus used since Sumerian and Babylonian times and still used today in many places around theworld is such a computer. High-confidence avionics software avoids the calculation of elusive radicals such as 2 p in their code for precisely this reason. We thank E. Feron for pointing this out. **More precisely, the roots cannot be expressed in terms of radicals. A pure mathematician might say “demoted”.