Research Summary

Adaptive Nonlinear Control. In the early 1990s Krstic developed the methods of adaptive control for nonlinear systems, the first tools for transient performance improvement in adaptive control, the first analysis of asymptotic behaviors in the absence of persistent excitation, the first proofs of robustnes of adaptive backstepping controllers to unmodeled dynamics, and the first adaptive controllers that have (inverse) optimality property over the entire infinite time interval. His PhD dissertation-based 1995 book [B1] with Kanellakopoulos and Kokotovic is the second most highly cited research book in control theory of all time.

Stochastic Nonlinear Control. In the late 1990s Krstic and his student Deng developed the first globally stabilizing controllers for stochastic nonlinear systems [B2]. He introduced the concept of noise-to-state stability (NSS) for systems whose noise covariance is unknown and arbitrary (including possibly being time-varying or state-dependent). He designed differential game controllers that achieve, with probability one, arbitrary peak-to-peak gain function assignment with respect to the noise covariance. These were the first stochastic nonlinear controllers given by explicit formulae.

Extremum Seeking. Working on control of gas turbine engine instabilities in the late 1990s, Krstic revived the classical "extremum seeking" method for real-time non-model based optimization from the 1950s, providing the first proof of its stability using nonlinear averaging theory and singular perturbation theory, developing compensators for convergence improvement, and extending extremum seeking to discrete time and to limit cycle minimization [B4]. As a result of Krstic’s advancements of extremum seeking methods, they have been adopted at a number of companies, including United Technologies (gas turbines, jet engine diffusers, and HVAC systems), Ford (engines), Northrop Grumman (endurance maximization for UAVs), Cymer (extreme ultraviolet light sources for photolithography), General Atomics (maglev trains), as well as by Los Alamos and Oak Ridge National Labs (charged particle accelerators), Livermore National Lab (engines), and by university researchers, including applications in photovoltaics, wind turbines, and aerodynamic flow control. 

Source Seeking in GPS-Denied Environments. Krstic introduced a methodology for using extremum seeking methods for solving source localization problems for autonomous and underactuated vehicles in GPS-denied environments, for application in environmental problems (contaminant tracking) and for steering fish-like vehicles using periodic forcing for both locomotion and position estimation. 

Stochastic Averaging and Stochastic Extremum Seeking. Motivated by biological “extremum seekers,” like E. Coli bacteria seeking food (chemotaxis), Krstic developed extremum seeking techniques that employ stochastic perturbations. To provide stability guarantees for these stochastic algorithms, he and his postdoc Liu developed major generalizations to the mathematical theory of stochastic averaging. 

Nash Equilibrium Seeking. Krstic extended extremum seeking from single-agent optimization to non-cooperative games. He proved, both for games with finitely many players and with uncountably many players, and when players employ either deterministic or stochastic extremum seeking algorithms, that the game converges to the underlying Nash equilibrium, despite the players not having modeling information on the payoff functions and not having information about the other players’ actions.

Control of Partial Differential Equations. Before Krstic, explicit control algorithms existed only for systems modeled by ODEs. For numerous applications involving fluids, elasticity, plasmas, and other spatially distributed dynamics modeled by PDEs, no methods existed for obtaining explicit formulae for feedback laws. The situation was even more hopeless for PDEs with nonlinearities or unknown parameters. Krstic and his student Smyshlyaev developed a radically new framework, called “continuum backstepping,” for converting PDE systems into desirable “target systems” using explicit transformations and feedback laws, which both employ Volterra-type integrals in spatial variables, with kernels resulting from solving linear PDEs of Goursat type on triangular domains. With this approach, Krstic was not only able to establish a general methodology for stabilizing linear PDEs [B6], but also for adaptive control of PDEs with unknown parameters [B8] and for nonlinear PDEs [J101]. Krstic’s method has seen application in problems whose complexity (nonlinear PDE models) makes them intractable by other methods, including estimation of spatial distribution of charge within Lithium-ion batteries and liquid cooling of large battery packs (with Bosch), control of flexible wings (with UIUC), control of current and kinetic profiles in fusion reactors (with General Atomics), control of automotive catalysts (with Ford), and control of problems in offshore oil production, including gas-liquid slugging flows in long pipes, stick-slip friction-induced instabilities in long flexible drilling systems, and underwater cable-based positioning systems.

Flow Control. Turbulent fluid flows are modeled by the notoriously difficult Navier-Stokes PDEs and remain a benchmark of difficulty for analysis and control design for PDEs. After developing controllers that induce mixing by feedback in low Reynolds number flows [B3] in channels, pipes, around bluff bodies, and for jet flows, Krstic and Vazquez achieved a breakthrough in developing control designs applicable to any Reynolds number, in 2D and 3D domains, and for flows that are electrically conducting and governed by a combination of Maxwell’s and Navier-Stokes equations (magnetohydrodynamic flows, such as plasmas and liquid metals used in cooling systems in fusion reactors) [B5]. In addition to stabilization, Krstic was the first to formulate and solve motion planning problems for Navier-Stokes systems, providing a method which, when developed for airfoil geometries, will enable the replacement of moving flap actuators by small fluidic actuators that trip the flow over the airfoil to generate desired waveforms of forces on the vehicle for its maneuvering, rather than changing the shape of the wing as moving flaps do.

Control of Systems with Long Delays. Applying his methods for PDEs to ODE systems with delays, in his sole-authored book [B7] Krstic developed results that revolutionize the field of delay systems. Krstic introduced a framework with nonlinear predictor operators, providing globally stabilizing nonlinear controllers in the presence of delays of arbitrary length and developing analysis methods for such nonlinear infinite-dimensional systems using continuum backstepping transformations. Krstic also developed controllers for delays that are rapidly time varying or highly uncertain. He and his student Bekiaris-Liberis solved the problem of stabilization of general nonlinear systems under state-dependent delays. 

Control of Sampled-Data Nonlinear Systems. Applying Krstic's predictor-based techniques to sampled-data nonlinear control systems, where only semiglobal practical stabilization under short sampling times was previously achievable, he and Karafyllis developed controllers that guarantee global stability under arbitrarily long sampling times. 

Other Applications and Industrial Collaborations. Krstic has also made contributions to control of rotating stall and surge in axial flow compressors of jet engines, thermoacoustic instabilities in gas turbine combustors, tuning of fuel injection waveforms for pulsed detonation engines, homogeneous charge compression ignition (HCCI) engines, wing rock instability in high angle-of-attack aircraft, helicopter blade-vortex interaction noise, bioreactor optimization, drag reduction in formation flight, control of mine ventilation networks, and, control of ramp-interconnected ships for cargo transfer in high sea states.