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Rudolf E. Kalman’s quest for algebraiccharacterizations of positivity

Tryphon T. Georgiou ∗

∗University of California, Irvine (e-mail: tryphon@uci.edu).

Abstract: The present paper is a tribute to Professor Rudolf Emil Kalman, father ofMathematical System Theory and a towering figure in the field of Control and DynamicalSystems. Amongst his seminal contributions was a series of results and insights into the roleof positivity in System Theory and in Control Engineering. The paper contains a collectionof reminiscences by the author together with brief technical references that touch upon theunparalleled influence of Professor Kalman on this topic.

Keywords: Rudolf E. Kalman

1. THE EARLY YEARS

The miracle years for Rudolf Kalman began in 1957,one year before he submitted his doctoral dissertationon the subject of sampled data systems. A constantstream of original ideas began pouring out, ideas thatwere to transform the field of control, educate and inspiregenerations of theorists, and enable transformative newtechnologies. Chief among those was the Kalman filter,published in Kalman [1960], a most impactful contributionthat is ubiquitous in today’s technological advances, fromguidance and navigation systems, cellphones, industrialcontrols to radar technology and vision systems. Butof no lesser importance were his insights and parallelcontributions to algebraic structure of dynamical systems,optimization, sampled data controls, and stability theory.The latter subject is intrinsically connected with positivityand passivity, concepts that were central to his thinkingthroughout his life.

His first paper on “Physical and mathematical mechanismsof instability in nonlinear automatic control,” published inKalman [1957], became very influential in the subsequentdevelopment of absolute stability theory (see Liberzon[2006]) and the topic never ceized to excite his interestand creativity. Shortly afterwards, in 1963, he publishedhis work on Lyapunov functions for the Lur’e problem,Kalman [1963] inspired by Yakubovich [1962]. This workbrought to life the so-called Kalman-Yacoubovich-Popov(KYP) lemma that highlighted the centrality of passivitytheory and was a harbinger of a great many developmentsthat took place in the subsequent years until the present(Iwasaki and Hara [2005], Rantzer [1996]). The KYPlemma proved the entry point for linear matrix inequalitiesand convex optimization in control theory (Willems [1971])and of key importance in filtering, stochastic realizations,control design, circuit theory as well as in addressingKalman’s far reaching question on the inverse regulatorproblem (Kalman [1964]).

Moving forward, Kalman explored this circle of ideas inproviding characterizations of stability with respect tomore general algebraic domains (Kalman [1965, 1969])

via conditions that capture positivity in a suitable sense.When reading these works, the converse question of char-acterizing domains via linear matrix inequalities is in-escapable.

2. POST 1970

A duality between quadratic regulator theory and stochas-tic realization theory, expanded upon and summarized inP. Faurre’s 1972 dissertation (cf. Faurre et al. [1978]),played a key role in Kalman’s thought during the fol-lowing years. Pursuing leads in early work by K. Lowner,Kalman sought to obtain an algebraic characterization ofminimal stochastic partial realizations (unpublished notesby Kalman). An alternative form of the same problemamounts to realizing a circuit with a minimal numberof passive elements (more accurately, minimal McMillandegree) based on the value of its impedance at specifiedcomplex frequencies. Potential leads were sought in invari-ants of linear quadratic problems – a subject of conver-sation between the late E. Bruce Lee and Kalman duringthe “1975 Birch Island Conference on Systems Theory” inBruce’s cabin (Figure 1).

The topic of “positive linear systems” remained a focusin his research proposals, motivated by applications tocircuits and modeling of random signals. However, para-phrasing his words, progress was frustratingly slow. Ina 1976 final progress report to AFOSR he highlightedthe apparent difficulties in dealing with the “interactionbetween positive and algebraic.” At that point, in thereport, he declared that “no further work on this topic isplanned in the near future.” But those who knew Kalman,knew better, that he never gives up. Soon afterwards in“Realization of covariance sequences,” Kalman [1982], hesought an answer in the form of algebraic inequalities interms of Schur parameters.

His insights pointed to computational algebraic geometrywhich, however, didn’t seem to have reached a satisfactorystage at the time for the sought purpose. The same seemedtrue for classical decision theory, in the style of Tarskiand Seidenberg. Under his direction, I began working

Fig. 1. E. Bruce Lee and Rudy Kalman at Birch Island in1975. Picture courtesy of E.B. Lee.

on my dissertation on this topic in 1981. Early on mydear friend Pramod Khargonekar joined me on this, andtogether we worked through the classical literature on themoment problem (Georgiou and Khargonekar [1982]). Yetthe “interaction between positive and algebraic” seemedonce again a roadblock towards the type of invariantsthat Kalman sought. One more year went by before I sawhow to characterize realizations of a fixed McMillan degreeby topological methods (see Georgiou [1983, 1987b,a]). Irecall with a mix of nostalgia my last years in Gainesvilleand all the ups and downs during the final stretch. Myinteractions with Kalman were points of inspiration as wellas contention, due to differing views on the nature of theproblem. While the sequel of Kalman’s problem on partialrealization of covariance sequences proved especially fruit-ful (see Byrnes et al. [1995], Byrnes and Lindquist [1997],Georgiou [1999], Byrnes et al. [2001, 2006]), Kalman’svision of an algebraic characterization was never fullyaccomplished. It is the sincere hope of the author thatthe present paper serves as a motivation and a challengeto others to attempt the next steps. Kalman’s dream andquest live on.

REFERENCES

Christopher I. Byrnes and Anders Lindquist. On dualitybetween filtering and interpolation. In Systems andControl in the Twenty-First Century, pages 101–136.Springer, 1997.

Christopher I. Byrnes, Anders Lindquist, Sergei V Gusev,and Alexei S Matveev. A complete parameterization ofall positive rational extensions of a covariance sequence.IEEE Transactions on Automatic Control, 40(11):1841–1857, 1995.

Christopher I. Byrnes, Tryphon T. Georgiou, and An-ders Lindquist. A generalized entropy criterion forNevanlinna-Pick interpolation with degree constraint.IEEE Transactions on Automatic Control, 46(6):822–839, 2001.

Christopher I. Byrnes, Tryphon T. Georgiou, AndersLindquist, and Alexander Megretski. Generalized in-terpolation in H∞ with a complexity constraint. Trans-actions of the American Mathematical Society, 358(3):965–987, 2006.

Fig. 2. R.E. Kalman with T.T. Georgiou at Kalman’s homein Gainesville, Florida on June 12, 2016. Picture takenby Mrs. Dina Kalman.

Pierre Faurre, Michel Clerget, and Francois Germain.Operateurs rationnels positifs. Dunod, Elsevier ScientificPublishing Co., 1978.

Tryphon T. Georgiou. Partial realization of covariancesequences. PhD thesis, University of Florida, 1983.

Tryphon T. Georgiou. Realization of power spectra frompartial covariance sequences. IEEE Transactions onAcoustics, Speech, and Signal Processing, 35(4):438–449,1987a.

Tryphon T. Georgiou. A topological approach toNevanlinna-Pick interpolation. SIAM journal on math-ematical analysis, 18(5):1248–1260, 1987b.

Tryphon T. Georgiou. The interpolation problem witha degree constraint. IEEE Transactions on AutomaticControl, 44(3):631–635, 1999.

Tryphon T. Georgiou and Pramod P. Khargonekar. Onthe partial realization problem for covariance sequences.In Proc. Princeton Conference on Information Systemsand Sciences, page 181, 1982.

Tetsuya Iwasaki and Shinji Hara. Generalized KYPlemma: unified frequency domain inequalities with de-sign applications. IEEE Transactions on AutomaticControl, 50(1):41–59, 2005.

Rudolf E. Kalman. When is a linear control systemoptimal? Journal of Basic Engineering, 86(1):51–60,1964.

Rudolf E. Kalman. On the Hermite-Fujiwara theorem instability theory. Quarterly of Applied Mathematics, 23(3):279–282, 1965.

Rudolf E. Kalman. Algebraic characterization of poly-nomials whose zeros lie in certain algebraic domains.Proceedings of the National Academy of Sciences, 64(3):818–823, 1969.

Rudolf E. Kalman. Realization of covariance sequences.In Toeplitz Centennial, pages 331–342. Springer, 1982.

Rudolph E. Kalman. Physical and mathematical mecha-nisms of instability in nonlinear automatic control sys-tems. Trans. ASME, 79(3):553–566, 1957.

Rudolph E. Kalman. A new approach to linear filteringand prediction problems. Journal of basic Engineering,82(1):35–45, 1960.

Rudolph E. Kalman. Lyapunov functions for the prob-lem of Lur’e in automatic control. Proceedings of thenational academy of sciences, 49(2):201–205, 1963.

MR Liberzon. Essays on the absolute stability theory. Au-tomation and remote control, 67(10):1610–1644, 2006.

Anders Rantzer. On the Kalman-Yakubovich-Popovlemma. Systems & Control Letters, 28(1):7–10, 1996.

Jan Willems. Least squares stationary optimal control andthe algebraic Riccati equation. IEEE Transactions onAutomatic Control, 16(6):621–634, 1971.

Vladimir A Yakubovich. The solution of certain matrixinequalities in automatic control theory. In Soviet Math.Dokl, volume 3, pages 620–623, 1962.