**Research
Summary**

Miroslav Krstic's research is in

◦
control
theory (nonlinear control, adaptive control, stochastic nonlinear systems, PDE
control, delay systems),

◦
model-free
optimization (extremum seeking),

◦
various
applications.

1. **Nonlinear
Control. **In the 1990s Krstic pioneered
feedback stabilization methods for nonlinear systems with unknown parameters
and stochastic disturbances.

**Adaptive Designs.** 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.

2. **PDE
Control.** 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. Around 2000 Krstic
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 and
his student Smyshlyaev were 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.

3. **Delay
Systems.** Applying his methods for PDEs to
ODE systems with arbitrarily long delays, in his sole-authored 2009 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.

**Sampled-Data Nonlinear Control.**** **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.

4. **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 ES. **Motivated
by biological “extremum seekers,” like E. Coli
bacteria seeking food (chemotaxis), Krstic developed extremum seeking
techniques that employ stochastic perturbations **[B9]**. 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.

5. **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.