Following two years of teaching at University of Belgrade, Krstić moved to the United States for graduate studies in 1991. He wrote his first journal paper a few weeks upon arrival,[4] with a solution that has transformed adaptive control.[5] He received his MSc in electrical engineering in 1992, and his PhD in 1995[3] (defended in December 1994), from University of California, Santa Barbara with Petar Kokotovic.[6]
Krstić's 455-page PhD dissertation[7] earned the campuswide Lancaster Best Dissertation Award at UC Santa Barbara and was published a few months later, with expansions, by John Wiley and Sons.[8][9]
Krstić received two Best Student Paper Awards. First, at the 1993 IEEE Conference on Decision and Control, for his paper on the nonlinear swapping approach to adaptive nonlinear control.[10] Second, at the 1996 American Control Conference,[11] for his single-authored paper on invariant manifolds in adaptive control.[12]
Additionally, for single-authored papers written while a PhD student, Krstić earned the O. Hugo Schuck[13] and George S. Axelby[14] outstanding paper awards.
Faculty career
After receiving his PhD in 1995, Krstić was an assistant professor at University of Maryland[3] for two years, In 1997, he was recruited as an associate professor at University of California, San Diego (UCSD),[3] and promoted to full professor three years later (2000).
Krstić’s grants from those five years include NSF Career, ONR YIP, and PECASE from President Clinton.
Since 2009, Krstić has held the Alspach endowed chair, and in 2015 was promoted to Distinguished Professor.[2]
In 2005, Krstić became the first engineering professor to receive the UC San Diego Chancellor's Award for Research,[15].
In 2008 Krstić founded the Cymer Center for Control Systems and Dynamics[16] and remains its director. Since 2012 he has served as Senior Associate Vice-Chancellor for Research at UCSD.[3][17]
Research in Control Theory
Krstić is a co-author of 18 books, about 480 journal papers,[2] and is the highest-published author in both of the flagship control systems journals, Automatica and IEEE Transactions on Automatic Control (according to Scopus[18]), with more than 100 papers in each of the two journals.[19][20]
NONLINEAR and ADAPTIVE CONTROL. Krstić’s 1995 book with Kanellakopoulos and Kokotovic,[9] an expanded version of his PhD dissertation,[7] pioneered adaptive stabilization methods for nonlinear systems with unknown parameters and is the second highest-cited control monograph.[21] (The top-cited control monograph is Boyd et al.[22]) Krstić introduced the tuning-function designs, modular designs, nonlinear swapping, passivity-based identifiers, adaptive CLFs and ISS-CLFs, and output-feedback adaptive nonlinear and linear controllers based on backstepping. STOCHASTIC STABILIZATION. Krstić and his student Deng[23] developed stabilizing controllers for stochastic nonlinear systems, introduced ISS-CLFs to stochastic systems, and designed differential game controllers that achieve, with probability one, peak-to-peak gain function assignment with respect to unknown noise covariance.
EXTREMUM SEEKING. Krstić pioneered, for general nonlinear dynamical systems, extremum seeking (ES) as an approach for real-time model-free optimization. To establish stability and performance guarantees, he introduced a combination of averaging and singular perturbation techniques to establish exponential stability.[24] Among Krstić’s advances of ES is source seeking for autonomous vehicles, model-free Nash equilibrium seeking for non-cooperative games, Newton-based ES for model-free assignment of convergence rate, model-free stabilization by minimum-seeking of CLFs, and ES for maps with large delays and PDEs with Oliveira.[25]STOCHASTIC AVERAGING AND STOCHASTIC EXTREMUM SEEKING. In introducing stochastic ES, Krstić and his postdoc Liu[26] generalized stochastic averaging theorems by removing the restrictions of global Lipschitzness, global exponential stability of the average system, vanishing noise, and finiteness of time horizon.
PDE BACKSTEPPING. Krstić generalized backstepping control from ODEs to PDEs. His IFAC Chestnut prize[27]-winning book[28] with his student Smyshlyaev provides an accessible introduction to PDE backstepping. PDE backstepping uses explicit and invertible Volterra-type integral transformations, with spatial integration kernels governed by linear PDEs of Goursat type on triangular domains. His general methodology stabilizes PDEs of parabolic and hyperbolic types, as well as of higher orders in space (Korteweg-De Vries, Schroedinger, Kuramoto-Sivashinsky, beams, etc.). For PDEs with unknown parameters, Krstić developed adaptive controllers.[29] Krstic applied PDE backstepping to traffic flows with his student Yu[30] and to additive manufacturing with his student Koga.[31]CONTROL OF NAVIER-STOKES SYSTEMS. For turbulent fluids, including electrically conducting flows with Maxwell’s PDEs (magnetohydrodynamic/MHD flows), Krstić and his student Vazquez developed flow control designs.[32]ISS FOR PDEs. Despite the unboundedness of input operators in PDEs with boundary inputs, Krstić and Karafyllis established ISS of PDEs, developed small-gain theorems for PDEs, and enabled analysis of interconnected PDEs from different classes.[33]
PREDICTORS FOR NONLINEAR DELAY SYSTEMS. In his single-authored 2009 Birkhäuser book[34] Krstić extended his hyperbolic PDE results to nonlinear ODEs with delays. He introduced nonlinear predictor operators of the infinite-dimensional delay state and launched control of interconnected PDE-ODE and PDE-PDE systems. In three subsequent books, Krstić and collaborators generalized the predictors to time- and state-dependent delays,[35] to delay-adaptive control for unknown delays,[36] and to sampled-data implementation.[37]
PRESCRIBED-TIME CONTROL. Krstić introduced time-varying techniques for the design and analysis of controllers[38] and observers[39] that achieve stabilization in user-prescribed time, independent of initial conditions, and even in the presence of deterministic disturbances and stochastic disturbances.[40]
SAFE, NON-OVERSHOOTING NONLINEAR CONTROL. In his 2006 paper,[41] Krstić pioneered a backstepping procedure for guaranteeing what is now referred to as "safety" and back then as "non-overshooting" control. He provided designs for CBFs of high relative degree and for achieving, for systems with unmatched disturbances, what is now referred to as input-to-state safety (ISSf). He extended deterministic non-overshooting control to nonlinear systems with stochastic disturbances.[42] He extended his prescribed-time (PT) idea from stabilization to safety, introducing PT safety filters,[43] which reduce the restrictiveness of conventional exponential safety filters. He converted his results on inverse optimal stabilization to inverse optimal safe control,[44] where a safety filter simultaneously maximizes safety and liveness, over the entire infinite time horizon. He generalized safety control from ODEs to PDEs.[45]
MACHINE LEARNING FOR PDE CONTROL. In his 2023 IEEE Bode Lecture,[46] Krstić introduced deep neural operators for off-line learning of PDE backstepping designs for hyperbolic and parabolic PDEs,[47][48] to enable online use of gain-scheduling backstepping for nonlinear PDEs and adaptive backstepping for PDEs with unknown parameters. He pioneered a roadmap for both the approximation theory for gain kernel PDEs and for stability guarantees under neural network approximations of the PDE backstepping controllers.
In control systems, Krstić is among the highest-cited researchers,[49][50][51][52] with a Google Scholar h-index over 125.[53] Among the living mechanical and aerospace engineers in the U.S., Krstić is the 8th highest cited according to Research.com[54] and Google Scholar.[55][56][57][58][53]
Krstić is Editor-in-Chief of Systems & Control Letters[59] and has been senior editor in Automatica[60] and IEEE Transactions on Automatic Control.[61]
CHIP PHOTOLITHOGRAPHY: Cymer Inc.,[62] a San Diego company with which Krstić co-founded the Cymer Center for Control Systems and Dynamics in 2008,[63] employed Krstić’s extremum seeking (ES) technology in 2012 to stabilize extreme ultraviolet (EUV) light sources. This boosted chip density 220-fold: from 193 nm resolution to 13 nm. Krstić’s 2002 discrete ES algorithm[64] is the basis of the seminal 2013 US Patent 8598552B1,[65] by his 4 ES trainees (Drs. Frihauf,[66] Riggs, Graham, Dunstan), who transitioned Krstić's ES technology as employees at Cymer. Shortly upon stabilizing EUV with extremum seeking, Cymer was acquired by ASML, for $3.7B.[67] EUV is a $10B/yr industry in 2024.[68] EUV is used by Intel,[69]IBM, Samsung, and TSMC.
AIRCRAFT CARRIERS: In 2014-2019, employed by General Atomics Electromagnetic Systems (San Diego) as a consultant, Krstić led his 4 former PhD trainees, hired by GA (Drs. G. Prior,[70] N. Ghods,[71] P. Frihauf,[72] C. Kinney[73]), in the control design and performance analysis for electromagnetic advanced arresting gear (AAG). Their controllers now manage all arrestments on the aircraft “supercarrier” USS Gerald R. Ford (CVN-78). In the US Navy 2020 video[74] the controllers by Krstić's team manage the landings of F/A-18 Super Hornet, C-2A Greyhound, and F-14 Tomcat aircraft. GA’s AAG is due to be also installed on USS Kennedy (CVN-79) and Enterprise (CVN-80).[75]
ACCELERATORS: Krstić and his students introduced the ES methodology to the field of charged particle accelerators.[76] His PhD graduate Dr. A. Scheinker implemented ES on the Los Alamos Lab's 1-km LANSCE accelerator, at several other U.S. Department of Energy labs (Lawrence Berkeley, Stanford Linear Accelerator, Argonne), and at other world-leading accelerators (CERN in Switzerland and Germany's Elektronen-Synchrotron DESY).[77] ES cuts accelerator re-tuning after upgrade from weeks to minutes.
MARS ROVER: The MS thesis[78] supervised by Krstić supplied the auto-focus algorithm for Mars Rover Curiosity’s ChemCam System, which performs chemical testing of Martian rocks.[79]
First engineering recipient of the UC San Diego Chancellor's Associates Award for Excellence in Science & Engineering Research (immediately following the Nobel laureate Roger Tsien)
For launching several new control system directions, Krstić has been recognized by International Federation of Automatic Control (IFAC) with a trifecta of triennial technical awards: IFAC TC Award on Non-Linear Control Systems,[95] IFAC TC Distributed Parameter Systems Ruth F. Curtain Award,[96] and IFAC TC Award on Adaptive and Learning Systems.[97][98] Each of the three areas is large, with a decades-long IFAC symposium series.[99][100][101] Krstić is the only researcher to receive such triennial awards for lifetime achievement in more than one controls area.
For his 50th birthday, as a tribute to his legacy, Krstić's colleagues published a monograph on nonlinear delay systems.[102]
Books
Nonlinear and Adaptive Control Design (1995), co-authored with Ioannis Kanellakopoulos and Petar Kokotovic; John Wiley and Sons. ISBN0-471-12732-9
Stabilization of Nonlinear Uncertain Systems (1998), co-authored with Hua Deng; Springer. ISBN1-85233-020-1
Flow Control by Feedback (2002), co-authored with Ole Morten Aamo; Springer. ISBN1-85233-669-2
Real-Time Optimization by Extremum Seeking Feedback (2003), co-authored with Kartik B. Ariyur; John Wiley and Sons. ISBN0-471-46859-2
Control of Turbulent and Magnetohydrodynamic Channel Flows (2007), co-authored with Rafael Vazquez; Birkhauser. ISBN978-0-8176-4698-1
Boundary Control of PDEs: A Course on Backstepping Designs (2008), co-authored with Andrey Smyshlyaev; SIAM. ISBN978-0-89871-650-4
Delay Compensation for Nonlinear, Adaptive, and PDE Systems (2009); Birkhauser. ISBN978-0-8176-4698-1
Adaptive Control of Parabolic PDEs (2010), co-authored with Andrey Smyshlyaev; Princeton University Press. ISBN978-0691142869
Stochastic Averaging and Stochastic Extremum Seeking (2012), co-authored with Shu-Jun Liu; Springer. ISBN978-1-4471-4086-3
Nonlinear Control Under Nonconstant Delays (2013), co-authored with Nikolaos Bekiaris-Liberis; SIAM. ISBN978-1-61197-284-9
Predictor Feedback for Delay Systems: Implementations and Approximations (2017), coauthored with Iasson Karafyllis; Birkhauser, ISBN978-3-319-42377-7
Model-Free Stabilization by Extremum Seeking (2017), co-authored with Alexander Scheinker; Springer. ISBN978-3-319-50790-3
Input-to-State Stability for PDEs (2018), co-authored with Iasson Karafyllis; Springer. ISBN978-3-319-91011-6
Delay-Adaptive Linear Control (2019), co-authored with Yang Zhu; Princeton University Press. ISBN9780691202549
Materials Phase Change PDE Control & Estimation: From Additive Manufacturing to Polar Ice (2020), co-authored with Shumon Koga; Springer. ISBN978-3-030-58490-0
PDE Control of String-Actuated Motion (2022); co-authored with Ji Wang, Princeton University Press. ISBN9780691233499
Extremum Seeking through Delays and PDEs (2022), co-authored with Tiago Roux Oliveira, SIAM. ISBN978-1-61197-734-9
Traffic Congestion Control by PDE Backstepping (2023), co-authored with Huan Yu, Birkhäuser. ISBN978-3-031-19345-3
for model-free assignment of convergence rate and for equalizing convergence across input channels of multivariable maps; with inversion of Hessian estimate using Riccati ODE