Miroslav Krstić

Education

Faculty career

Research in Control Theory

1. NONLINEAR and ADAPTIVE CONTROL. Krstić’s 1995 book with Kanellakopoulos and Kokotovic, an expanded version of his PhD dissertation, pioneered adaptive stabilization methods for nonlinear systems with unknown parameters and is the second highest-cited control monograph. 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 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.
2. EXTREMUM SEEKING.  Krstić pioneered, for general nonlinear dynamical systems, extremum seeking 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. 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.  STOCHASTIC AVERAGING AND STOCHASTIC EXTREMUM SEEKING. In introducing stochastic ES, Krstić and his postdoc Liu 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.
3. PDE BACKSTEPPING. Krstić generalized backstepping control from ODEs to PDEs. His IFAC Chestnut prize-winning book 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. For PDEs with unknown parameters, Krstić developed adaptive controllers.  Krstic applied PDE backstepping to traffic flows with his student Yu and to additive manufacturing with his student Koga. CONTROL OF NAVIER-STOKES SYSTEMS. For turbulent fluids, including electrically conducting flows with Maxwell’s PDEs, Krstić and his student Vazquez developed flow control designs. 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.
4. PREDICTORS FOR NONLINEAR DELAY SYSTEMS. In his single-authored 2009 Birkhäuser book 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, to delay-adaptive control for unknown delays, and to sampled-data implementation.
5. PRESCRIBED-TIME CONTROL. Krstić introduced time-varying techniques for the design and analysis of controllers and observers that achieve stabilization in user-prescribed time, independent of initial conditions, and even in the presence of deterministic disturbances and stochastic disturbances.
6. SAFE, NON-OVERSHOOTING NONLINEAR CONTROL. In his 2006 paper, 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. He extended deterministic non-overshooting control to nonlinear systems with stochastic disturbances. He extended his prescribed-time idea from stabilization to safety, introducing PT safety filters,' which reduce the restrictiveness of conventional exponential safety filters. He converted his results on inverse optimal stabilization to inverse optimal safe control, where a safety filter simultaneously maximizes safety and liveness, over the entire infinite time horizon. He generalized safety control from ODEs to PDEs.
7. MACHINE LEARNING FOR PDE CONTROL. In his 2023 IEEE Bode Lecture, Krstić introduced deep neural operators for off-line learning of PDE backstepping designs for hyperbolic and parabolic PDEs,' 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.

Industry Work

Awards

• IEEE Roger W. Brockett Control Systems Award
• French National Centre for Scientific Research Fellow-Ambassador
• IEEE Hendrik W. Bode Lecture Prize
• AACC Richard E. Bellman Control Heritage Award
• ASME Rufus Oldenburger Medal
• SIAM W. T. and Idalia Reid Prize
• A. V. "Bal" Balakrishnan
• AACC John R. Ragazzini Award
• IFAC
• IFAC TC Award on Non-Linear Control Systems
• IFAC TC Distributed Parameter Systems Ruth F. Curtain Award
• IFAC TC Award on Adaptive and Learning Systems
• IFAC TC Delay Systems "Lifetime Achievement" Award
• ASME
• ASME

• First engineering recipient of the UC San Diego Chancellor's Associates Award for Excellence in Science & Engineering Research
• Presidential Early Career Award for Scientists and Engineers
• Office of Naval Research Young Investigator Award
• National Science Foundation Career Award

Books

1. Nonlinear and Adaptive Control Design, co-authored with Ioannis Kanellakopoulos and Petar Kokotovic; John Wiley and Sons.
2. Stabilization of Nonlinear Uncertain Systems, co-authored with Hua Deng; Springer.
3. Flow Control by Feedback, co-authored with Ole Morten Aamo; Springer.
4. Real-Time Optimization by Extremum Seeking Feedback, co-authored with Kartik B. Ariyur; John Wiley and Sons.
5. Control of Turbulent and Magnetohydrodynamic Channel Flows, co-authored with Rafael Vazquez; Birkhauser.
6. Boundary Control of PDEs: A Course on Backstepping Designs, co-authored with Andrey Smyshlyaev; SIAM.
7. Delay Compensation for Nonlinear, Adaptive, and PDE Systems ; Birkhauser.
8. Adaptive Control of Parabolic PDEs, co-authored with Andrey Smyshlyaev; Princeton University Press.
9. Stochastic Averaging and Stochastic Extremum Seeking, co-authored with Shu-Jun Liu; Springer.
10. Nonlinear Control Under Nonconstant Delays, co-authored with Nikolaos Bekiaris-Liberis; SIAM.
11. Predictor Feedback for Delay Systems: Implementations and Approximations, coauthored with Iasson Karafyllis; Birkhauser,
12. Model-Free Stabilization by Extremum Seeking, co-authored with Alexander Scheinker; Springer.
13. Input-to-State Stability for PDEs, co-authored with Iasson Karafyllis; Springer.
14. Delay-Adaptive Linear Control, co-authored with Yang Zhu; Princeton University Press.
15. Materials Phase Change PDE Control & Estimation: From Additive Manufacturing to Polar Ice, co-authored with Shumon Koga; Springer.
16. PDE Control of String-Actuated Motion ; co-authored with Ji Wang, Princeton University Press.
17. Extremum Seeking through Delays and PDEs, co-authored with Tiago Roux Oliveira, SIAM.
18. Traffic Congestion Control by PDE Backstepping, co-authored with Huan Yu, Birkhäuser.
19. Robust Adaptive Control: Deadzone-Adapted Disturbance Suppression, co-authored with Iasson Karafyllis,

Methods pioneered by Krstić

Adaptive Nonlinear Control

1. tuning-function design
2. * adaptive backstepping with a single parameter estimator, for unmatched parametric uncertainties
3. modular designs
4. * combine any parameter estimator with any ISS controller
5. nonlinear swapping
6. * stability analysis with nonlinear filter-based gradient and least-squares parameter estimators for nonlinear systems
7. passivity-based identifiers
8. * identifiers with observers and nonlinear damping
9. adaptive CLFs and ISS-CLFs
10. * general frameworks for Lyapunov and ISS-based adaptive nonlinear control
11. output-feedback nonlinear and linear adaptive backstepping
12. * adaptive observer-based controllers with K-filters

Stochastic Nonlinear Stabilization

1. stochastic backstepping
2. * backstepping employing Itô calculus for continuous-time stochastic systems; ensures stability in probability
3. noise-to-state stability
4. * ISS in probability with respect to unknown covariance of noise
5. inverse optimal differential games w.r.t. noise covariance
6. * optimal assignment of integral gains from covariance to state

Extremum Seeking

1. stability of extremum seeking for general nonlinear dynamical systems
2. * analysis via singular perturbations and averaging of reduced model
3. source seeking
4. * search/navigation in space for GPS-denied autonomous vehicles and robots
5. Nash equilibrium seeking
6. * ES in non-cooperative game multi-agent setting
7. Newton-based ES
8. * 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
9. model-free stabilization with ES
10. * by minimum-seeking of CLFs
11. ES for maps with large delays and PDEs
12. * employing PDE backstepping
13. stochastic extremum seeking
14. * for ES with random walk perturbations, like employed by E.Coli bacteria
15. generalized stochastic averaging
16. * without restrictions

PDE Backstepping

1. backstepping transformations, kernel PDEs
2. * transformations into desirable target PDEs
3. * analysis of Goursat-form PDEs for gain kernels
4. backstepping for parabolic and hyperbolic PDEs
5. * designs of full-state stabilizing feedback law and convergent observers
6. backstepping for PDE-ODE and PDE-PDE cascades
7. * for cascades like ODEs with parabolic PDE input dynamic and reaction-diffusion PDEs with input delays
8. adaptive PDE backstepping
9. * for PDEs with unknown functional parameters, using Lyapunov, swapping, and passive estimators
10. traffic flow stabilization
11. * control of ARZ PDEs
12. additive manufacturing
13. * control of Stefan PDEs

Navier-Stokes Stabilization and Mixing

1. Backstepping for turbulent flows and MHD systems
2. * Stabilizing controllers and observers at high Reynolds and Hartmann numbers
3. ''Fluid mixing by optimal de-stabilization''

ISS for PDEs

1. ISS to boundary inputs
2. small-gain theorems for PDEs
3. * for parabolic and hyperbolic PDEs

Predictors for Nonlinear Delay Systems

1. nonlinear predictors
2. * including predictors for state-dependent delays
3. delay-adaptive control
4. * for linear systems with unknown delays and other plant parameters
5. approximate nonlinear predictors
6. * for real-time and sampled-data implementation

Prescribed-Time Control

1. PT stabilization
2. * backstepping for driving state to setpoint by user-prescribed time regardless of initial condition
3. PT observers
4. * for state estimate convergence in arbitrary finite time
5. stochastic PT control
6. * for PT stabilization in probability

Non-overshooting and Safe Nonlinear Control

1. high relative degree CBFs
2. * recursive backstepping design of CBFs without shrinking the safe set
3. ISSf gain assignment
4. * backstepping design of non-overshooting controllers that assign a desired input-to-state safety gain function
5. stochastic non-overshooting control
6. * to guarantee safety at least in the mean
7. prescribed-time safety filters
8. * to let the state reach the safety boundary by the time the prohibition on the unsafe set is lifted
9. inverse optimal safety filters
10. * safety filters with safety-maximization and liveness-maximization along entire time horizon, under deterministic and stochastic disturbances
11. safe control for PDEs
12. * for Stefan, liquid-tank, gas-piston, and chemostat PDEs

Deep Neural Operators for PDE Control

1. universal approximability theory for backstepping kernel PDEs
2. stability guarantees under ML approximation of PDE backstepping
3. * for hyperbolic and parabolic PDEs