arXiv subjects

Miroslav Krstić

Publications and source records attributed to Miroslav Krstić.

26 records · Page 2Linked to original sources

arXiv2024

Extremum and Nash Equilibrium Seeking with Delays and PDEs: Designs & Applications

The development of extremum seeking (ES) has progressed, over the past hundred years, from static maps, to finite-dimensional dynamic systems, to networks of static and dynamic agents. Extensions from ODE dynamics to maps and agents that incorporate delays or even partial differential equations (PDEs) is the next natural step in that progression through ascending research challenges. This paper reviews results on algorithm design and theory of ES for such infinite-dimensional systems. Both hyperbolic and parabolic dynamics are presented: delays or transport equations, heat-dominated equation, wave equations, and reaction-advection-diffusion equations. Nash equilibrium seeking (NES) methods are introduced for noncooperative game scenarios of the model-free kind and then specialized to single-agent optimization. Even heterogeneous PDE games, such as a duopoly with one parabolic and one hyperbolic agent, are considered. Several engineering applications are touched upon for illustration, including flow-traffic control for urban mobility, oil-drilling systems, deep-sea cable-actuated source seeking, additive manufacturing modeled by the Stefan PDE, biological reactors, light-source seeking with flexible-beam structures, and neuromuscular electrical stimulation.

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arXiv2024

Extremum Seeking is Stable for Scalar Maps that are Strictly but Not Strongly Convex

For a map that is strictly but not strongly convex, model-based gradient extremum seeking has an eigenvalue of zero at the extremum, i.e., it fails at exponential convergence. Interestingly, perturbation-based model-free extremum seeking has a negative Jacobian, in the average, meaning that its (practical) convergence is exponential, even though the map's Hessian is zero at the extremum. While these observations for the gradient algorithm are not trivial, we focus in this paper on an even more nontrivial study of the same phenomenon for Newton-based extremum seeking control (NESC). NESC is a second-order method which corrects for the unknown Hessian of the unknown map, not only in order to speed up parameter convergence, but also (1) to make the convergence rate user-assignable in spite of the unknown Hessian, and (2) to equalize the convergence rates in different directions for multivariable maps. Previous NESC work established stability only for maps whose Hessians are strictly positive definite everywhere, so the Hessian is invertible everywhere. For a scalar map, we establish the rather unexpected property that, even when the map behind is strictly convex but not strongly convex, i.e., when the Hessian may be zero, NESC guarantees practical asymptotic stability, semiglobally. While a model-based Newton-based algorithm would run into non-invertibility of the Hessian, the perturbation-based NESC, surprisingly, avoids this challenge by leveraging the fact that the average of the perturbation-based Hessian estimate is always positive, even though the actual Hessian may be zero.

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arXiv2024

Backstepping Neural Operators for $2\times 2$ Hyperbolic PDEs

Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more gain kernels-a PDE structure unaddressed thus far with DeepONet. In this paper, we explore the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants by considering a simple counter-convecting $2\times 2$ coupled system in whose control a $2\times 2$ kernel PDE system in Goursat form arises. Engineering applications include oil drilling, the Saint-Venant model of shallow water waves, and the Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. We establish the continuity of the mapping from a total of five plant PDE functional coefficients to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and ensure that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our $L^2$-Globally-exponentially stabilizing (GES) approximate gain kernel-based output feedback design implies the deep learning of both the controller's and the observer's gains. Moreover, the encoding of the output-feedback law into DeepONet ensures semi-global practical exponential stability (SG-PES). The DeepONet operator speeds up the computation of the controller gains by multiple orders of magnitude. Its theoretically proven stabilizing capability is demonstrated through simulations.

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arXiv2024

Sliding-Mode Nash Equilibrium Seeking for a Quadratic Duopoly Game

This paper introduces a new method to achieve stable convergence to Nash equilibrium in duopoly noncooperative games. Inspired by the recent fixed-time Nash Equilibrium seeking (NES) as well as prescribed-time extremum seeking (ES) and source seeking schemes, our approach employs a distributed sliding mode control (SMC) scheme, integrating extremum seeking with sinusoidal perturbation signals to estimate the pseudogradients of quadratic payoff functions. Notably, this is the first attempt to address noncooperative games without relying on models, combining classical extremum seeking with relay components instead of proportional control laws. We prove finite-time convergence of the closed-loop average system to Nash equilibrium using stability analysis techniques such as time-scaling, Lyapunov's direct method, and averaging theory for discontinuous systems. Additionally, we quantify the size of residual sets around the Nash equilibrium and validate our theoretical results through simulations.

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arXiv2024

Nash Equilibrium Seeking for Noncooperative Duopoly Games via Event-Triggered Control

This paper proposes a novel approach for locally stable convergence to Nash equilibrium in duopoly noncooperative games based on a distributed event-triggered control scheme. The proposed approach employs extremum seeking, with sinusoidal perturbation signals applied to estimate the Gradient (first derivative) of unknown quadratic payoff functions. This is the first instance of noncooperative games being tackled in a model-free fashion integrated with the event-triggered methodology. Each player evaluates independently the deviation between the corresponding current state variable and its last broadcasted value to update the player action, while they preserve control performance under limited bandwidth of the actuation paths and still guarantee stability for the closed-loop dynamics. In particular, the stability analysis is carried out using time-scaling technique, Lyapunov's direct method and averaging theory for discontinuous systems. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an example.

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arXiv2023

Deep Learning of Delay-Compensated Backstepping for Reaction-Diffusion PDEs

Deep neural networks that approximate nonlinear function-to-function mappings, i.e., operators, which are called DeepONet, have been demonstrated in recent articles to be capable of encoding entire PDE control methodologies, such as backstepping, so that, for each new functional coefficient of a PDE plant, the backstepping gains are obtained through a simple function evaluation. These initial results have been limited to single PDEs from a given class, approximating the solutions of only single-PDE operators for the gain kernels. In this paper we expand this framework to the approximation of multiple (cascaded) nonlinear operators. Multiple operators arise in the control of PDE systems from distinct PDE classes, such as the system in this paper: a reaction-diffusion plant, which is a parabolic PDE, with input delay, which is a hyperbolic PDE. The DeepONet-approximated nonlinear operator is a cascade/composition of the operators defined by one hyperbolic PDE of the Goursat form and one parabolic PDE on a rectangle, both of which are bilinear in their input functions and not explicitly solvable. For the delay-compensated PDE backstepping controller, which employs the learned control operator, namely, the approximated gain kernel, we guarantee exponential stability in the $L^2$ norm of the plant state and the $H^1$ norm of the input delay state. Simulations illustrate the contributed theory.

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arXiv2023

Stabilization of Age-Structured Chemostat Hyperbolic PDE with Actuator Dynamics

For population systems modeled by age-structured hyperbolic partial differential equations (PDEs), we redesign the existing feedback laws, designed under the assumption that the dilution input is directly actuated, to the more realistic case where dilution is governed by actuation dynamics (modeled simply by an integrator). In addition to the standard constraint that the population density must remain positive, the dilution dynamics introduce constraints of not only positivity of dilution, but possibly of given positive lower and upper bounds on dilution. We present several designs, of varying complexity, and with various measurement requirements, which not only ensure global asymptotic (and local exponential) stabilization of a desired positive population density profile from all positive initial conditions, but do so without violating the constraints on the dilution state. To develop the results, we exploit the relation between first-order hyperbolic PDEs and an equivalent representation in which a scalar input-driven mode is decoupled from input-free infinite-dimensional internal dynamics represented by an integral delay system.

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arXiv2022

Underactuated Source Seeking by Surge Force Tuning: Theory and Boat Experiments

We extend source seeking algorithms, in the absence of position and velocity measurements, and with tuning of the surge input, from velocity-actuated (unicycle) kinematic models to force-actuated generic Euler-Lagrange dynamic underactuated models. In the design and analysis, we employ a symmetric product approximation, averaging, passivity, and partial-state stability theory. The proposed control law requires only real-time measurement of the source signal at the current position of the vehicle and ensures semi-global practical uniform asymptotic stability (SPUAS) with respect to the linear motion coordinates for the closed-loop system. The performance of our source seeker with surge force tuning is illustrated with both numerical simulations and experiments of an underactuated boat.

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