Miroslav Krstic – Research Summary

An abbreviated version of this narrative (one-page, less technical, no references) is available at http://flyingv.ucsd.edu/krstic/research-sum.html

Clicking on a blue item in the narrative will display the paper in question from Krstic’s web server

1. Nonlinear Adaptive Control. Following the ground-breaking achievements in geometric control theory in the 1980s, the needs in applications turned the focus to the questions of robustness to uncertainties in the systems’ vector fields. In the early 1990s Krstic pioneered feedback stabilization methods for nonlinear systems with unknown parameters. He developed a comprehensive theoretical arsenal of Lyapunov-based [J3, J5, J9] and non-Lyapunov-based methods [J6, J8, J11] for design and analysis of adaptive controllers for nonlinear systems. He provided a method that removes overparameterization that plagued previous adaptive nonlinear control designs [J3], necessary and sufficient clf-based conditions for adaptive stabilization [J9, J11] in the spirit of Sontag’s results for non-adaptive stabilization, the first method for systematic transient performance improvement in adaptive control [J4], the first analysis of asymptotic behaviors in the absence of persistent excitation [J13], the first proofs of robustness of adaptive backstepping controllers to unmodeled dynamics [J12, J20, J22], the first adaptive controllers that are guaranteed to remain stabilizing even when adaptation is turned off [J7], and the first adaptive controllers that have (inverse) optimality property over the entire infinite time interval [J19, J111]. He received four best paper awards for his work in this period, including the Axelby Prize for the best paper in the IEEE Transactions on Automatic Control for the single-authored paper [J13], where he quantifies the measure of initial conditions from which adaptive controllers may asymptotically approach destabilizing values of the controller gains by completely characterizing the topology of invariant manifolds in the closed-loop adaptive system. 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, with over 4700 citations as of September 2012.

2. Stochastic Stabilization. In the late 1990s Krstic and his student Deng developed the first globally stabilizing controllers for stochastic nonlinear systems [B2]. He provided necessary and sufficient clf-based conditions for stabilization of stochastic systems [J17, J18, J45] and provided full-state and output-feedback designs [J25, J33] for stochastic stabilization. Extending his deterministic results on ISS-clfs for inverse optimal disturbance attenuation [J19], he introduced the concept of noise-to-state stability (NSS) for systems whose noise covariance is unknown and time-varying (which is a proper stochastic equivalent of Sontag’s ISS for systems with random inputs) and designed differential game controllers that achieve, with probability one, arbitrary peak-to-peak gain function assignment with respect to the noise covariance [J45]. He developed the first LaSalle-type theorem free of globally Liptschitz restrictions [J45], which allowed him to introduce globally stabilizing feedback laws for stochastic nonlinear systems with unknown constant covariance. This body of work provided explicit feedback laws for stochastic nonlinear systems, without requiring solutions of Hamilton-Jacobi PDEs.

3. PDE Control. Before Krstic’s work in the area of control of partial differential equations, explicit control algorithms existed only for systems modeled by ODEs and for elementary cases of open-loop stable PDEs where exponential stabilization is achieved by simple proportional feedback, such as by “boundary dampers.” For numerous classes of unstable PDEs and for numerous applications involving fluids, elasticity, plasmas, and other spatially distributed dynamics, 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 able to establish a general methodology for stabilizing linear PDEs, which is presented in a tutorial format in the SIAM-published graduate textbook [B5], which was a finalist for IFAC’s Harold Chestnut Textbook Prize.

Krstic has developed stabilizing feedback laws for a vast array of PDE classes, including parabolic [J73, J76, J78] and hyperbolic classes (of the first [J96] and second order [J92, J130]), Burgers [J28, J40, J42], Korteweg-de Vries [J41], Kuramoto-Sivashinsky [J36], Schrodinger [J149, J155], Ginzburg-Landau [J71, J86], thermal convection [J77, J99], wave equations with unconventional anti-stiffness [J92] and anti-damping [J122], Euler beams [J118], shear beams [J93], Timoshenko beams [J116], as well as cascades of ODEs with parabolic [J119, J143] and hyperbolic [J120, J140] PDEs.

For nonlinear PDEs of a very general parabolic class, which includes semilinear PDEs without a restriction on the nonlinearity growth, Krstic and Vazquez introduced a general design based on spatial Volterra series operators [J101, J102] whose kernels are found by solving a sequence of linear PDEs of Goursat type on domains whose dimension grows to infinity but whose volumes decays to zero, allowing them to prove convergence of the control laws and stability in closed loop. This is at present the most general design approach available for stabilizing nonlinear PDEs that are not structurally restricted or open-loop stable. For the special case of a viscous Burgers equation, Krstic finds the nonlinear feedback operators explicitly [J108] and solves the motion planning explicitly using the backstepping approach [J109].

Inspired by Krener’s backstepping observers for nonlinear ODEs, Krstic developed their PDE equivalents for various PDEs whose state is measurable only at the boundary [J76, J92, J98, J99, J109]. He also established the separation principle—stability of full-state feedback control laws employing the estimates from the PDE observers.

For PDEs with unknown parameters, such as viscosity in parabolic PDEs, advection speed in transport PDEs, or damping in wave PDEs, Krstic developed a series of adaptive control designs [J47, J88, J89, J90, J123, J133, J138], made possible by the explicit form of the controllers arising from the backstepping approach. This work has culminated by the book [B8] that presents an array of approaches for system identification and adaptive control for PDEs of parabolic type.

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 and stick-slip friction-induced instabilities in long flexible drilling systems (Mines Paris-Tech). His approach to PDE control has even led to control laws for deployment of multi-agent systems into geometric shapes of previously unattained generality [J146, J152].

4. Control of Navier-Stokes Systems. 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 stabilization or mixing (depending on the gain sign) by feedback in low Reynolds number flows [B3] in channels [J48, J60], pipes [J68], around bluff bodies, and for jet flows, Krstic and Vazquez achieved a breakthrough [B6] in developing control designs applicable to any Reynolds number, in 2D and 3D domains [J91, J117], for flows that are electrically conducting and governed by a combination of Maxwell’s and Navier-Stokes equations [J100] (magnetohydrodynamic flows, such as plasmas and liquid metals used in cooling systems in fusion reactors), and for flows that are measurable only on the boundary [J98]. In addition to stabilization, Krstic was the first to formulate and solve motion planning problems for Navier-Stokes systems [J113], 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.

5. 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 control of delay systems. Krstic first employed his analysis method based on backstepping transformations to answer long-standing question of robustness of predictor feedbacks to delay mismatch [J107], the question of stability of predictor feedbacks for rapidly time-varying delays left open in the 1982 work of Artstein [J125], and to design delay-adaptive controllers for highly uncertain delays [J123, J138]. He then 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 [J121]. He and his student Bekiaris-Liberis solved the problems of stabilization of general nonlinear systems under time-varying delays [J154] and state-dependent delays [J162, J163], as well as for linear systems with distributed delays [J142, J166]. 

Inspired by Datko’s 1988 famed examples that controllers for certain PDE systems possess zero robustness margin to delays, Krstic developed control laws that compensate delays of any length in feedback designs for parabolic [J126] and second-order hyperbolic [J136] PDEs. He and Wang also characterized all the delay values that do not induce instability when applying the simple boundary damper feedback to wave PDEs [J144].

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 [J156, J159]. 

6. Stochastic Averaging. Motivated by biological “gradient climbers” like nutrient-seeking E. Coli bacteria, Krstic developed stochastic extremum seeking algorithms that represent plausible simple feedback laws executed by individual bacteria when performing “chemotaxis” [B9]. To provide stability guarantees for such stochastic algorithms, he and his postdoc Liu developed major generalizations to the mathematical theory of stochastic averaging and stability. They considered continuous-time nonlinear systems with stochastic perturbations and developed theorems on stochastic averaging that remove the long-standing restrictions of global Lipschitzness of the vector field, global exponential stability of the average system, equilibrium preservation under perturbation, and the finiteness of the time interval [J131]. They further relax the condition of uniform convergence of the stochastic perturbation in [J134, J150] and employ the resulting theorems in stochastic extremum seeking algorithms for cooperative and non-cooperative optimization. In [J135] they address bacterial chemotaxis and prove that each bacterium, modeled as a nonholonomic unicycle and applying stochastic extremum seeking through its steering input, achieves exponential convergence to the area of maximum nutrient concentration. Krstic and Krieger employed the new stochastic averaging theory in developing algorithms to maximize the time a UAV can remain airborne on a tank of fuel by tuning the airspeed of the UAV with the help of atmospheric turbulence acting as a perturbation for a stochastic extremum seeking algorithm [J153].

7. Extremum Seeking. Working on control of gas turbine engine instabilities in the late 1990s [J23, J32, J39], Krstic revived the classical early-1950s-era "extremum seeking" method for real-time non-model based optimization [B4]. He provided the first proof of its stability using nonlinear averaging theory and singular perturbation theory [J35], developed compensators for convergence improvement and for maps that evolve with time according to an exosystem [J37], extended extremum seeking to discrete time [J49] and to limit cycle minimization [J34], and introduced recently a generalized Newton-type extremum seeking algorithm whose convergence is not only guaranteed but also user-assignable [J161].

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 (laser pulse shaping and extreme ultraviolet light sources for photolithography), General Atomics (maglev trains, tokamak fusion reactors, and novel modular fission reactors), as well as by Los Alamos and Oak Ridge National Labs (charged particle accelerators), Livermore National Lab (engines), with him playing part in a few of these transitions [J56, J94, J112, J153, J158]. Many university-affiliated practitioners have also adopted extremum seeking, using it in novel applications in photovoltaics, wind turbines, and aerodynamic flow control. 

In joint work with his student Paul Frihauf and Tamer Basar on Nash equilibrium seeking, Krstic extended extremum seeking from single-agent optimization to non-cooperative games [J157]. They proved, both for games with finitely many players and with uncountably many players, 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.

8. Nonholonomic Source Seeking. While nonholonomic vehicles violate the requirement of exponential stability of the plant in the original extremum seeking approach, Krstic modified the approach to make it applicable to solving source localization problems for autonomous and underactuated vehicles in GPS-denied environments. He and his students developed algorithms that can tune either the longitudinal velocity [J84] or angular velocity [J104, 129, 145], and that can seek sources not only in 2D but also in 3D [J115]. He also provided a plausible mathematical explanation of how fish track prey using only the sense of smell and simple extremum seeking algorithms [J124]. By considering models of nonholonomic kinematics of fish in potential or vortex flows, developed by Marsden and others in the early 2000s, he showed that the periodic forcing (tail flapping) that fish use for locomotion can be modulated in a simple manner using the measured scent of the prey to achieve prey tracking without position measurement. In other words, he took the topic of fish locomotion from the question of what open-loop signals generate particular gaits to what feedback laws fish use to steer themselves in the dark.