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Stability theory plays a crucial role in feedback control. However, adaptive control theory requires advanced and specialized stability notions that are not frequently used in standard feedback control theory. The present document is a set of notes for a graduate course. It describes the global stability notions needed in (robust) adaptive control and develops the mathematical tools that are used for the proof of such stability properties. Moreover, the document shows why and how these global stability properties arise in adaptive control. We focus on stability properties for time-invariant systems. Consequently, tracking control problems are not covered by the present document., Comment: 48 pages, 4 figures

Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable methodologies, for such a significant portion of real systems, have been historically scarce. Around the year 2000, the backstepping approach to PDE control began to offer not only a less abstract alternative to PDE control techniques replicating optimal and spectrum assignment techniques of the 1960s, but also enabled the methodologies of adaptive and nonlinear control, matured in the 1980s and 1990s, to be extended from ODEs to PDEs, allowing feedback synthesis for physical and engineering systems that are uncertain, nonlinear, and infinite-dimensional. The PDE backstepping literature has grown in its nearly a quarter century of development to many hundreds of papers and nearly a dozen books. This survey aims to facilitate the entry, for a new researcher, into this thriving area of overwhelming size and topical diversity. Designs of controllers and observers, for parabolic, hyperbolic, and other classes of PDEs, in one and more dimensions (in box and spherical geometries), with nonlinear, adaptive, sampled-data, and event-triggered extensions, are covered in the survey. The lifeblood of control are technology and physics. The survey places a particular emphasis on applications that have motivated the development of the theory and which have benefited from the theory and designs: applications involving flows, flexible structures, materials, thermal and chemically reacting dynamics, energy (from oil drilling to batteries and magnetic confinement fusions), and vehicles., Comment: Preprint submitted to Automatica

Populations do not only interact over time but also age over time. It is therefore common to model them as age-structured PDEs, where age is the space variable. Since the models also involve integrals over age, both in the birth process and in the interaction among species, they are in fact integro-partial differential equations (IPDEs) with positive states. To regulate the population densities to desired profiles, harvesting is used as input. But non-discriminating harvesting, where wanting to repress one species will inevitably repress the other species as well, the positivity restriction on the input (no insertion of population), and the multiplicative nature of harvesting, makes control challenging even for ODE versions of such dynamics, let alone for their IPDE versions on an infinite-dimensional nonnegative state space. We introduce a design for a benchmark version of such a problem: a two-population predator-prey setup. The model is equivalent to two coupled ordinary differential equations (ODEs), actuated by harvesting which must not drop below zero, and strongly disturbed by two autonomous but exponentially stable integral delay equations (IDEs). We develop two control designs. With a modified Volterra-like control Lyapunov function, we design a simple feedback which employs possibly negative harvesting for global stabilization of the ODE model, while guaranteeing regional regulation with positive harvesting. With a more sophisticated, restrained controller we achieve regulation for the ODE model globally, with positive harvesting. For the full IPDE model, with the IDE dynamics acting as large disturbances, for both the simple and saturated feedback laws we provide explicit estimates of the regions of attraction. The paper charts a new pathway for control designs for infinite-dimensional multi-species dynamics and for nonlinear positive systems with positive controls., Comment: submitted to IEEE Transactions on Automatic Control

Fixed-time stable dynamical systems are capable of achieving exact convergence to an equilibrium point within a fixed time that is independent of the initial conditions of the system. This property makes them highly appealing for designing control, estimation, and optimization algorithms in applications with stringent performance requirements. However, the set of tools available for analyzing the interconnection of fixed-time stable systems is rather limited compared to their asymptotic counterparts. In this paper, we address some of these limitations by exploiting the emergence of multiple time scales in nonlinear singularly perturbed dynamical systems, where the fast dynamics and the slow dynamics are fixed-time stable on their own. By extending the so-called composite Lyapunov method from asymptotic stability to the context of fixed-time stability, we provide a novel class of Lyapunov-based sufficient conditions to certify fixed-time stability in a class of singularly perturbed dynamical systems. The results are illustrated, analytically and numerically, using a fixed-time gradient flow system interconnected with a fixed-time plant and an additional high-order example.

We employ the recent performance-barrier event-triggered control (P-ETC) for achieving global exponential convergence of a class of reaction-diffusion PDEs via PDE backstepping control. Rather than insisting on a strictly monotonic decrease of the Lyapunov function for the closed-loop system, P-ETC allows the Lyapunov function to increase as long as it remains below an acceptable performance-barrier. This approach integrates a performance residual, the difference between the value of the performance-barrier and the Lyapunov function, into the triggering mechanism. The integration adds flexibility and results in fewer control updates than with regular ETC (R-ETC) that demands a monotonic decrease of the Lyapunov function. Our P-ETC PDE backstepping design ensures global exponential convergence of the closed-loop system in the spatial L^2 norm, without encountering Zeno phenomenon. To avoid continuous monitoring of the triggering function that generates events, we develop periodic event-triggered and self-triggered variants (P-PETC and P-STC, respectively) of the P-ETC. The P-PETC only requires periodic evaluation of the triggering function whereas the P-STC preemptively computes the time of the next event at the current event time using the system model and continuously available system states. The P-PETC and P-STC also ensure a Zeno-free behavior and deliver performance equivalent to that of the continuous-time P-ETC which requires continuous evaluation of the triggering function, in addition to the continuous sensing of the state. We provide numerical simulations to illustrate the proposed technique and to compare it with R-ETC associated with strictly decreasing Lyapunov functions.

In socio-technical multi-agent systems, deception exploits privileged information to induce false beliefs in "victims," keeping them oblivious and leading to outcomes detrimental to them or advantageous to the deceiver. We consider model-free Nash-equilibrium-seeking for non-cooperative games with asymmetric information and introduce model-free deceptive algorithms with stability guarantees. In the simplest algorithm, the deceiver includes in his action policy the victim's exploration signal, with an amplitude tuned by an integrator of the regulation error between the deceiver's actual and desired payoff. The integral feedback drives the deceiver's payoff to the payoff's reference value, while the victim is led to adopt a suboptimal action, at which the pseudogradient of the deceiver's payoff is zero. The deceiver's and victim's actions turn out to constitute a "deceptive" Nash equilibrium of a different game, whose structure is managed - in real time - by the deceiver. We examine quadratic, aggregative, and more general games and provide conditions for a successful deception, mutual and benevolent deception, and immunity to deception. Stability results are established using techniques based on averaging and singular perturbations. Among the examples in the paper is a microeconomic duopoly in which the deceiver induces in the victim a belief that the buyers disfavor the deceiver more than they actually do, leading the victim to increase the price above the Nash price, and resulting in an increased profit for the deceiver and a decreased profit for the victim. A study of the deceiver's integral feedback for the desired profit reveals that, in duopolies with equal marginal costs, a deceiver that is greedy for very high profit can attain any such profit, and pursue this with arbitrarily high integral gain (impatiently), irrespective of the market preference for the victim.

Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction-diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45x relative to the traditional finite difference solvers for every timestep in the simulation trajectory., Comment: 13 pages, 4 figures

This paper deals with the gradient extremum seeking control for static scalar maps with actuators governed by distributed diffusion partial differential equations (PDEs). To achieve the real-time optimization objective, we design a compensation controller for the distributed diffusion PDE via backstepping transformation in infinite dimensions. A further contribution of this paper is the appropriate motion planning design of the so-called probing (or perturbation) signal, which is more involved than in the non-distributed counterpart. Hence, with these two design ingredients, we provide an averaging-based methodology that can be implemented using the gradient and Hessian estimates. Local exponential stability for the closed-loop equilibrium of the average error dynamics is guaranteed through a Lyapunov-based analysis. By employing the averaging theory for infinite-dimensional systems, we prove that the trajectory converges to a small neighborhood surrounding the optimal point. The effectiveness of the proposed extremum seeking controller for distributed diffusion PDEs in cascade of nonlinear maps to be optimized is illustrated by means of numerical simulations., Comment: 8 pages and 7 figures

This paper presents a complete inverse optimality study for a linearized tank-liquid system where the liquid is described by the viscous Saint-Venant model with surface tension and possible wall friction. We define an appropriate weak solution notion for which we establish existence/uniqueness results with inputs that do not necessarily satisfy any compatibility condition as well as stabilization results with feedback laws that are constructed with the help of a Control Lyapunov Functional. We show that the proposed family of stabilizing feedback laws is optimal for a certain meaningful quadratic cost functional. Finally, we show that the optimal feedback law guarantees additional stronger stability estimates which are similar to those obtained in the case of classical solutions., Comment: 40 pages

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., Comment: 8 pages and 2 figures. arXiv admin note: substantial text overlap with arXiv:2404.07287

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., Comment: 6 pages, 5 figures

Over the last decade, data-driven methods have surged in popularity, emerging as valuable tools for control theory. As such, neural network approximations of control feedback laws, system dynamics, and even Lyapunov functions have attracted growing attention. With the ascent of learning based control, the need for accurate, fast, and easy-to-use benchmarks has increased. In this work, we present the first learning-based environment for boundary control of PDEs. In our benchmark, we introduce three foundational PDE problems - a 1D transport PDE, a 1D reaction-diffusion PDE, and a 2D Navier-Stokes PDE - whose solvers are bundled in an user-friendly reinforcement learning gym. With this gym, we then present the first set of model-free, reinforcement learning algorithms for solving this series of benchmark problems, achieving stability, although at a higher cost compared to model-based PDE backstepping. With the set of benchmark environments and detailed examples, this work significantly lowers the barrier to entry for learning-based PDE control - a topic largely unexplored by the data-driven control community. The entire benchmark is available on Github along with detailed documentation and the presented reinforcement learning models are open sourced., Comment: 26 pages 10 figures. Accepted L4DC 2024

This article provides an overview of the design of nonlinear feedback Cruise Controllers (CCs) for automated vehicles on lane-free roads. The feedback design problem is particularly challenging because of the various state constraints (e.g., collision-free movement, road geometry, speed limits) as well as the nature of the control objective (globally stabilizing distributed controllers that require measurements from neighboring vehicles only). Therefore, the resulting nonlinear control system is defined on an open set (not necessarily diffeomorphic to a linear space) for which the set of desired equilibria is non-compact. The proposed design of the CCs is based on energy-like control Lyapunov functions which combine potential functions with kinetic energy terms and other appropriate penalty terms. The feedback design in the microscopic level is accompanied by the derivation of the corresponding macroscopic traffic flow models. Explicit relations are established between selectable CC features and the obtained macroscopic traffic flow characteristics. This facilitates the active design of efficient traffic flow with desired properties, i.e., the construction of artificial traffic fluids., Comment: 53 pages

Exploring novel strategies for the regulation of axon growth, we introduce a periodic event-triggered control (PETC) to enhance the practical implementation of the associated PDE backstepping control law. Neurological injuries may impair neuronal function, but therapies like Chondroitinase ABC (ChABC) have shown promise in improving axon elongation by influencing the extracellular matrix. This matrix, composed of extracellular macromolecules and minerals, regulates tubulin protein concentration, potentially aiding in neuronal recovery. The concentration and spatial distribution of tubulin influence axon elongation dynamics. Recent research explores feedback control strategies for this model, leading to the development of an event-triggering control (CETC) approach. In this approach, the control law updates when the monitored triggering condition is met, reducing actuation resource consumption. Through the meticulous redesign of the triggering mechanism, we introduce a periodic event-triggering control (PETC), updating control inputs at specific intervals, but evaluating the event-trigger only periodically, an ideal tool for standard time-sliced actuators like ChABC. PETC is a step forward to the design of practically feasible feedback laws for the neuron growth process. The PETC strategy establishes an upper bound on event triggers between periodic examinations, ensuring convergence and preventing Zeno behavior. Through Lyapunov analysis, we demonstrate the local exponential convergence of the system with the periodic event-triggering mechanism in the $L^2$-norm sense. Numerical examples are presented to confirm the theoretical findings., Comment: Submitted to 2024 Conference on Decision and Control

We present Assignably Safe Extremum Seeking (ASfES), an algorithm designed to minimize a measured objective function while maintaining a measured metric of safety (a control barrier function or CBF) be positive in a practical sense. We ensure that for trajectories with safe initial conditions, the violation of safety can be made arbitrarily small with appropriately chosen design constants. We also guarantee an assignable ``attractivity'' rate: from unsafe initial conditions, the trajectories approach the safe set, in the sense of the measured CBF, at a rate no slower than a user-assigned rate. Similarly, from safe initial conditions, the trajectories approach the unsafe set, in the sense of the CBF, no faster than the assigned attractivity rate. The feature of assignable attractivity is not present in the semiglobal version of safe extremum seeking, where the semiglobality of convergence is achieved by slowing the adaptation. We also demonstrate local convergence of the parameter to a neighborhood of the minimum of the objective function constrained to the safe set. The ASfES algorithm and analysis are multivariable, but we also extend the algorithm to a Newton-Based ASfES scheme (NB-ASfES) which we show is only useful in the scalar case. The proven properties of the designs are illustrated through simulation examples.

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.

We consider the problem of inverse optimal control design for systems that are not affine in the control. In particular, we consider some classes of partial differential equations (PDEs) with quadratic convection and counter-convection, for which the L2 norm is a control Lyapunov function (CLF) whose derivative has either a depressed cubic or a quadratic dependence in the boundary control input. We also consider diffusive PDEs with or without linear convection, for which a weighted L2 norm is a CLF whose derivative has a quadratic dependence in the control input. For each structure on the derivative of the CLF, we achieve inverse optimality with respect to a meaningful cost functional. For the case where the derivative of the CLF has a depressed cubic dependence in the control, we construct a cost functional for which the unique minimizer is the unique real root of a cubic polynomial: the Cardano-Lyapunov controller. When the derivative of the CLF is quadratic in the control, we construct a cost functional that is minimized by two distinct feedback laws, that correspond to the two distinct real roots of a quadratic equation. We show how to switch from one root to the other to reduce the control effort., Comment: To Appear in Proceedings of the 2024 European Control Conference

In this work we advance the recently-introduced deep learning-powered approach to PDE backstepping control by proposing a method that approximates only the control gain function -- a function of one variable -- instead of the entire kernel function of the backstepping transformation, which depends on two variables. This idea is introduced using several benchmark unstable PDEs, including hyperbolic and parabolic types, and extended to 2X2 hyperbolic systems. By employing a backstepping transformation that utilizes the exact kernel (suitable for gain scheduling) rather than an approximated one (suitable for adaptive control), we alter the quantification of the approximation error. This leads to a significant simplification in the target system, shifting the perturbation due to approximation from the domain to the boundary condition. Despite the notable differences in the Lyapunov analysis, we are able to retain stability guarantees with this simplified approximation approach. Approximating only the control gain function simplifies the operator being approximated and the training of its neural approximation, potentially reducing the neural network size. The trade-off for these simplifications is a more intricate Lyapunov analysis, involving higher Sobolev spaces for some PDEs, and certain restrictions on initial conditions arising from these spaces. It is crucial to carefully consider the specific requirements and constraints of each problem to determine the most suitable approach; indeed, recent works have demonstrated successful applications of both full-kernel and gain-only approaches in adaptive control and gain scheduling contexts., Comment: Preprint submitted to CAM

We propose the first generalization of Sontag s universal controller to systems not affine in the control, particularly, to PDEs with boundary actuation. We assume that the system admits a control Lyapunov function (CLF) whose derivative, rather than being affine in the control, has either a depressed cubic, quadratic, or depressed quartic dependence on the control. For each case, a continuous universal controller that vanishes at the origin and achieves global exponential stability is derived. We prove our result in the context of convectionreaction-diffusion PDEs with Dirichlet actuation. We show that if the convection has a certain structure, then the L2 norm of the state is a CLF. In addition to generalizing Sontag s formula to some non-affine systems, we present the first general Lyapunov approach for boundary control of nonlinear PDEs. We illustrate our results via a numerical example., Comment: To be presented at the 2024 American Control Conference

There have been recent efforts that combine seemingly disparate methods, extremum seeking (ES) optimization and partial differential equation (PDE) backstepping, to address the problem of model-free optimization with PDE actuator dynamics. In contrast to prior PDE-compensating ES designs, which only guarantee local stability around the extremum, we introduce unbiased ES that compensates for delay and diffusion PDE dynamics while ensuring exponential and unbiased convergence to the optimum. Our method leverages exponentially decaying/growing signals within the modulation/demodulation stages and carefully selected design parameters. The stability analysis of our designs relies on a state transformation, infinite-dimensional averaging, local exponential stability of the averaged system, local stability of the transformed system, and local exponential stability of the original system. Numerical simulations are presented to demonstrate the efficacy of the developed designs., Comment: Submitted to the 63rd IEEE Conference on Decision and Control (CDC), 2024

Our recently developed "unbiased" extremum seeking (uES) algorithms ensure perfect convergence to the optimum at a user-assigned exponential rate or, more powerfully, within a user-prescribed time. Unlike classical approach, these algorithms use time-varying adaptation and controller gains, along with constant or time-varying probing frequencies (chirp signals). This paper advances our earlier uES designs from strongly convex maps with static optima to a broader class of convex cost functions with time-varying optima diverging at arbitrary rates, even in finite time. This advancement first motivates the use of Lie bracket averaging instead of classical averaging due to the average system system, which doesn't necessarily need to be exponentially convergent, and the existence of non-periodic time-varying parameters; second, it necessitates the formulation of non-trivial and key feasibility conditions for the choice of time-varying design parameters and their decay/growth rates in relation to the convexity of the map and the divergence rate of optima. These conditions indicate that, for constant-frequency probing, the user-defined asymptotic rate of unbiasing is limited by the convexity of the map. However, this rate can be made arbitrarily fast (including asymptotic, exponential, and prescribed time) using chirpy probing, which requires sufficiently rapid frequency and adaptation growth to enable tracking of faster-diverging optima. In addition to numerical simulations of the designs, we experimentally test the feasibility of exponential uES for tuning the angular velocity of a unicycle to seek a static light source., Comment: Submitted to Transactions on Automatic Control

In this paper, we extend our previous work on the power series method for computing backstepping kernels. Our first contribution is the development of initial steps towards a MATLAB toolbox dedicated to backstepping kernel computation. This toolbox would exploit MATLAB's linear algebra and sparse matrix manipulation features for enhanced efficiency; our initial findings show considerable improvements in computational speed with respect to the use of symbolical software without loss of precision at high orders. Additionally, we tackle limitations observed in our earlier work, such as slow convergence (due to oscillatory behaviors) and non-converging series (due to loss of analiticity at some singular points). To overcome these challenges, we introduce a technique that mitigates this behaviour by computing the expansion at different points, denoted as localized power series. This approach effectively navigates around singularities, and can also accelerates convergence by using more local approximations. Basic examples are provided to demonstrate these enhancements. Although this research is still ongoing, the significant potential and simplicity of the method already establish the power series approach as a viable and versatile solution for solving backstepping kernel equations, benefiting both novel and experienced practitioners in the field. We anticipate that these developments will be particularly beneficial in training the recently introduced neural operators that approximate backstepping kernels and gains., Comment: Preprint submitted to CDC 2024

Nonlinear convection, the source of turbulence in fluid flows, may hold the key to stabilizing turbulence by solving a specific cubic polynomial equation. We consider the incompressible Navier-Stokes equations in a two-dimensional channel. The tangential and normal velocities are assumed to be periodic in the streamwise direction. The pressure difference between the left and right ends of the channel is constant. Moreover, we consider no-slip boundary conditions, that is, zero tangential velocity, at the top and bottom walls of the channel, and normal velocity actuation at the top and bottom walls. We design the boundary control inputs to achieve global exponential stabilization, in the L2 sense, of a chosen Poiseuille equilibrium profile for an arbitrarily large Reynolds number. The key idea behind our approach is to select the boundary controllers such that they have zero spatial mean (to guarantee mass conservation) but non-zero spatial cubic mean. We reveal that, because of convection, the time derivative of the L2 energy of the regulation error is a cubic polynomial in the cubic mean of the boundary inputs. Regulation is then achieved by solving a specific cubic equation, using the Cardano root formula. The results are illustrated via a numerical example., Comment: To be presented at the 63rd IEEE Conference on Decision and Control (CDC 2024)

In this paper we extend our recently proposed Deadzone-Adapted Disturbance Suppression (DADS) Control approach from systems with matched uncertainties to general systems in parametric strict feedback form. The DADS approach prevents gain and state drift regardless of the size of the disturbance and unknown parameter and achieves an attenuation of the plant output to an assignable small level, despite the presence of persistent disturbances and unknown parameters of arbitrary and unknown bounds. The controller is designed by means of a step-by-step backstepping procedure which can be applied in an algorithmic fashion. Examples are provided which illustrate the efficiency of the DADS controller compared to existing adaptive control schemes., Comment: 32 pages, 6 figures. arXiv admin note: text overlap with arXiv:2311.07938

This paper introduces extremum seeking (ES) algorithms designed to achieve perfect tracking of arbitrary time-varying extremum. In contrast to classical ES approaches that employ constant frequencies and controller gains, our algorithms leverage time-varying parameters, growing either asymptotically or exponentially, to achieve desired convergence behaviors. Our stability analysis involves state transformation, time-dilation transformation, and Lie bracket averaging. The state transformation is based on the multiplication of the input state by asymptotic or exponential growth functions. The time transformation enables tracking of the extremum as it gradually converges to a constant value when viewed in the dilated time domain. Finally, Lie bracket averaging is applied to the transformed system, ensuring practical uniform stability in the dilated time domain as well as asymptotic or exponential stability of the original system in the original time domain. We validate the feasibility of these designs through numerical simulations., Comment: Accepted to the 2024 American Control Conference (ACC)

Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the initial state estimate at the beginning of the horizon as well as the measured output and input signals over the moving time horizon. In this paper we introduce MHEs for PDEs which remove the need for a numerical solution of an observer PDE in real time. We accomplish this using the PDE backstepping method which, for certain classes of both hyperbolic and parabolic PDEs, produces moving-horizon state estimates explicitly. Precisely, to explicitly produce the state estimates, we employ a backstepping transformation of a hard-to-solve observer PDE into a target observer PDE, which is explicitly solvable. The MHEs we propose are not new observer designs but simply the explicit MHE realizations, over a moving horizon of arbitrary length, of the existing backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose as duals of MPC, but they are given explicitly, even for PDEs. In the paper we provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as well as simulation results that illustrate theoretically guaranteed convergence of the MHEs., Comment: 7 pages, 1 figure, submitted to ACC 2024

To stabilize PDE models, control laws require space-dependent functional gains mapped by nonlinear operators from the PDE functional coefficients. When a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent, a gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. The GS version of PDE backstepping employs gains obtained by solving a PDE at each value of the state. Performing such PDE computations in real time may be prohibitive. The recently introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value, without requiring a PDE solution. In this paper we introduce NOs for GS-PDE backstepping. GS controllers act on the premise that the state change is slow and, as a result, guarantee only local stability, even for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation using both a "full-kernel" approach and the "gain-only" approach to gain operator approximation. Numerical simulations illustrate stabilization and demonstrate speedup by three orders of magnitude over traditional PDE gain-scheduling. Code (Github) for the numerical implementation is published to enable exploration., Comment: 16 pages, 5 figures

We present multivariable extremum seeking (ES) designs that achieve unbiased convergence to the optimum. Two designs are introduced: one with exponential unbiased convergence (unbiased extremum seeker, uES) and the other with user-assignable prescribed-time unbiased convergence (unbiased PT extremum seeker, uPT-ES). In contrast to the conventional ES, which uses persistent sinusoids and results in steady-state oscillations around the optimum, the exponential uES employs an exponentially decaying amplitude in the perturbation signal (for achieving convergence) and an exponentially growing demodulation signal (for making the convergence unbiased). The achievement of unbiased convergence also entails employing an adaptation gain that is sufficiently large in relation to the decay rate of the perturbation amplitude. Stated concisely, the bias is eliminated by having the learning process outpace the waning of the perturbation. The other algorithm, uPT-ES, employs prescribed-time convergent/blow-up functions in place of constant amplitudes of sinusoids, and it also replaces constant-frequency sinusoids with chirp signals whose frequency grows over time. Among the convergence results in the ES literature, uPT-ES may be the strongest yet in terms of the convergence rate (prescribed-time) and accuracy (unbiased). To enhance the robustness of uES to a time-varying optimum, exponential functions are modified to keep oscillations at steady state. Stability analysis of the designs is based on a state transformation, averaging, local exponential/PT stability of the averaged system, local stability of the transformed system, and local exponential/PT stability of the original system. For numerical implementation of the developed ES schemes and comparison with previous ES designs, the problem of source seeking by a two-dimensional velocity-actuated point mass is considered., Comment: 16 pages, 7 figures

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.

This paper proposes an event-triggered control scheme for multivariable extremum seeking of static maps. Both static and dynamic triggering conditions are developed. Integrating Lyapunov and averaging theories for discontinuous systems, a systematic design procedure and stability analysis are developed. Both event-based methods enable one to achieve an asymptotic stability result. Ultimately, the resulting closed-loop dynamics demonstrates the advantages of combining both approaches, namely, event-triggered control and extremum seeking. Although we keep the presentation using the classical event-triggered method, the extension of the results for the periodic event-triggered approach is also indicated. An illustration of the benefits of the new control method is presented using consistent simulation results, which compare the static and the dynamic triggering approaches., Comment: 21 pages, 6 figures, and 1 table

In this paper we study a special class of systems: time-invariant control systems that satisfy the matching condition for which no bounds for the disturbance and the unknown parameters are known. For this class of systems, we provide a simple, direct, adaptive control scheme that combines three elements: (a) nonlinear damping, (b) single-gain adjustment, and (c) deadzone in the update law. It is the first time that these three tools are combined and the proposed controller is called a Deadzone-Adapted Disturbance Suppression (DADS) Controller. The proposed adaptive control scheme achieves for the first time an attenuation of the plant state to an assignable small level, despite the presence of disturbances and unknown parameters of arbitrary and unknown bounds. Moreover, the DADS Controller prevents gain and state drift regardless of the size of the disturbance and unknown parameter., Comment: 26 pages, 3 figures

This paper presents the design and analysis of the extremum seeking for static maps with input passed through a partial differential equation (PDE) of the diffusion type defined on a time-varying spatial domain whose boundary position is governed by an ordinary differential equation (ODE). This is the first effort to pursue an extension of extremum seeking from the heat PDE to the Stefan PDE. We compensate the average-based actuation dynamics by a controller via backstepping transformation for the moving boundary, which is utilized to transform the original coupled PDE-ODE into a target system whose exponential stability of the average equilibrium of the average system is proved. The discussion for the delay-compensated extremum seeking control of the Stefan problem is also presented and illustrated with numerical simulations., Comment: 10 pages and 10 figures

In this paper, we investigate the rapid stabilization of N-layer Timoshenko composite beams with anti-damping and anti-stiffness at the uncontrolled boundaries. The problem of stabilization for a two-layer composite beam has been previously studied by transforming the model into a 1-D hyperbolic PIDE-ODE form and then applying backstepping to this new system. In principle this approach is generalizable to any number of layers. However, when some of the layers have the same physical properties (as e.g. in lamination of repeated layers), the approach leads to isotachic hyperbolic PDEs (i.e. where some states have the same transport speed). This particular yet physical and interesting case has not received much attention beyond a few remarks in the early hyperbolic design. Thus, this work starts by extending the theory of backstepping control of (m + n) hyperbolic PIDEs and m ODEs to blocks of isotachic states, leading to a block backstepping design. Then, returning to multilayer Timoshenko beams, the Riemann transformation is used to transform the states of N-layer Timoshenko beams into a 1-D hyperbolic PIDE-ODE system. The block backstepping method is then applied to this model, obtaining closed-loop stability of the origin in the L2 sense. An arbitrarily rapid convergence rate can be obtained by adjusting control parameters. Finally, numerical simulations are presented corroborating the theoretical developments., Comment: Preprint submitted to Automatica

We introduce a dynamic event-triggering mechanism for regulating the axonal growth of a neuron. We apply boundary actuation at the soma (the part of a neuron that contains the nucleus) and regulate the dynamics of tubulin concentration and axon length. The control law is formulated by applying a Zero-Order Hold (ZOH) to a continuous-time controller which guides the axon to reach the desired length. The proposed dynamic event-triggering mechanism determines the specific time instants at which control inputs are sampled from the continuous-time control law. We establish the existence of a minimum dwell-time between two triggering times that ensures avoidance of Zeno behavior. Through employing the Lyapunov analysis with PDE backstepping, we prove the local stability of the closed-loop system in $L_2$-norm, initially for the target system, and subsequently for the original system. The effectiveness of the proposed method is showcased through numerical simulations., Comment: Accepted to American Control Conference 2024

We introduce a type of safe extremum seeking (ES) controller, which minimizes an unknown objective function while also maintaining practical positivity of an unknown barrier function. We show semi-global practical asymptotic stability of our algorithm and present an analogous notion of practical safety. The dynamics of the controller are inspired by the quadratic program (QP) based safety filter designs which, in the literature, are more commonly used in cases where the barrier function is known. Conditions on the barrier and objective function are explored showing that non convex problems can be solved. A Lyapunov argument is proposed to achieve the main results of the paper. Finally, an example is given of the algorithm which solves the constrained optimization problem.

Adaptive safe control employing conventional continuous infinite-time adaptation requires that the initial conditions be restricted to a subset of the safe set due to parametric uncertainty, where the safe set is shrunk in inverse proportion to the adaptation gain. The recent regulation-triggered adaptive control approach with batch least-squares identification (BaLSI, pronounced "ballsy") completes perfect parameter identification in finite time and offers a previously unforeseen advantage in adaptive safe control, which we elucidate in this paper. Since the true challenge of safe control is exhibited for CBF of a high relative degree, we undertake a safe BaLSI design in this paper for a class of systems that possess a particularly extreme relative degree: ODE-PDE-ODE sandwich systems. Such sandwich systems arise in various applications, including delivery UAVs (Unmanned Aerial Vehicles) with a cable-suspended load. Collision avoidance of the payload with the surrounding environment is required. The considered class of plants is $2\times2$ hyperbolic PDEs sandwiched by a strict-feedback nonlinear ODE and a linear ODE, where the unknown coefficients, whose bounds are known and arbitrary, are associated with the PDE in-domain coupling terms that can cause instability and with the input signal of the distal ODE. This is the first safe adaptive control design for PDEs, where we introduce the concept of PDE CBF whose non-negativity, as well as the ODE CBF's non-negativity, are ensured with a backstepping-based safety filter. Our safe adaptive controller is explicit and operates in the entire original safe set.

We demonstrate the recent designs of Safe Extremum Seeking (Safe ES) on the 1 kilometer-long charged particle accelerator at the Los Alamos Neutron Science Center (LANSCE). Safe ES is a modification of ES which, in addition to minimizing an analytically unknown cost, also employs a safety filter based on an analytically unknown control barrier function (CBF) safety metric. Accelerator tuning is necessitated by the accelerators being large, with many drifting parameters due to thermal effects and degradation. At the same time, safe operation (the maintenance of state constraints) is crucial, as damage brings astronomical costs, both financially and in operation downtime. Our measured (but analytically unknown) safety metric is the beam current. We perform multivariable Safe ES on three accelerator applications, in which we adapt 4, 6, and 3 magnet strength parameters, respectively. Two of the three applications are for validated simulation models of beamlines at LANSCE: the first for the Proton Radiography (pRad) beamline of 800 MeV protons for spot size tuning; the second on a high performance code, HPSim, for tuning the low energy beam transport (LEBT) region of of 750 keV protons. The third is an experimental tuning of the steering magnets in the LEBT at LANSCE.

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.

For the quite extensively developed PDE backstepping methodology for coupled linear hyperbolic PDEs, we provide a generalization from finite collections of such PDEs, whose states at each location in space are vector-valued, to previously unstudied infinite (continuum) ensembles of such hyperbolic PDEs, whose states are function-valued. The motivation for studying such systems comes from traffic applications (where driver and vehicle characteristics are continuously parametrized), fluid and structural applications, and future applications in population dynamics, including epidemiology. Our design is of an exponentially stabilizing scalar-valued control law for a PDE system in two independent dimensions, one spatial dimension and one ensemble dimension. In the process of generalizing PDE backstepping from finite to infinite collections of PDE systems, we generalize the results for PDE backstepping kernels to the continuously parametrized Goursat-form PDEs that govern such continuously parametrized kernels. The theory is illustrated with a simulation example, which is selected so that the kernels are explicitly solvable, to lend clarity and interpretability to the simulation results., Comment: 16 pages, 4 figures, to be published

The topics of source seeking and Newton-based extremum seeking have flourished, independently, but never combined. We present the first Newton-based source seeking algorithm. The algorithm employs forward velocity tuning, as in the very first source seeker for the unicycle, and incorporates an additional Riccati filter for inverting the Hessian inverse and feeding it into the demodulation signal. Using second-order Lie bracket averaging, we prove convergence to the source at a rate that is independent of the unknown Hessian of the map. The result is semiglobal and practical, for a map that is quadratic in the distance from the source. The paper presents a theory and simulations, which show advantage of the Newton-based over the gradient-based source seeking.

The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output or input. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator, mapping functions on a spatial domain into functions on a spatial domain, and where this gain-generating operator's inputs are the PDE's coefficients. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight. Once we produce this approximation-theoretic result in infinite dimension, with it we establish stability in closed loop under feedback that employs approximate gains. In addition to supplying such results under full-state feedback, we also develop DeepONet-approximated observers and output-feedback laws and prove their own stabilizing properties under neural operator approximations. With numerical simulations we illustrate the theoretical results and quantify the numerical effort savings, which are of two orders of magnitude, thanks to replacing the numerical PDE solving with the DeepONet., Comment: 20 pages

We develop a delay-adaptive controller for a class of first-order hyperbolic partial integro-differential equations (PIDEs) with an unknown input delay. By employing a transport PDE to represent delayed actuator states, the system is transformed into a transport partial differential equation (PDE) with unknown propagation speed cascaded with a PIDE. A parameter update law is designed using a Lyapunov argument and the infinite-dimensional backstepping technique to establish global stability results. Furthermore, the well-posedness of the closed-loop system is analyzed. Finally, the effectiveness of the proposed method was validated through numerical simulations

A static non-linear homogeneous feedback for a fixed-time stabilization of a linear time-invariant (LTI) system is designed in such a way that the settling time is assigned exactly to a prescribed constant for all nonzero initial conditions. The constant convergence time is achieved due to a dependence of the feedback gain of the initial state of the system. The robustness of the closed-loop system with respect to measurement noises and exogenous perturbations is studied using the concept of Input-to-State Stability (ISS). Both delay-free and input delay systems are studied. Theoretical results are illustrated by numerical simulations.

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.

Flow control occupies a special place in the fields of partial differential equations (PDEs) and control theory, where the complex behavior of solutions of nonlinear dynamics in very high dimension is not just to be understood but also to be assigned specific desired properties, by feedback control. Among several benchmark problems in flow control, the liquid-tank problem is particularly attractive as a research topic. In the liquid-tank problem the objective is to move a tank filled with liquid, suppress the nonlinear oscillations of the liquid in the process, bring the tank and liquid to rest, and avoid liquid spillage in the process. In other words, this is a problem of nonlinear PDE stabilization subject to state constraints. This review article focuses only on recent results on liquid-tank stabilization for viscous liquids. All possible cases are studied: with and without friction from the tank walls, with and without surface tension. Moreover, novel results are provided for the linearization of the tank-liquid system. The linearization of the tank-liquid system gives a high-order PDE which is a combination of a wave equation with Kelvin-Voigt damping and an Euler-Bernoulli beam equation. The feedback design methodology presented in the article is based on Control Lyapunov Functionals (CLFs), suitably extended from the CLF methodology for ODEs to the infinite-dimensional case. The CLFs proposed are modifications and augmentations of the total energy functionals for the tank-liquid system, so that the dissipative effects of viscosity, friction, and surface tension are captured and additional dissipation by feedback is made relatively easy. The article closes with an extensive list of open problems., Comment: 40 pages, submitted for possible publication

A nonovershooting finite-time control design for linear multi-input system is proposed by upgrading a linear (asymptotic) nonovershooting stabilizer to a homogeneous one. Robustness of the safety and stability properties is analyzed using the concept of Input-to-State Stability (ISS) on invariant sets and Input-to-State Safety (ISSf). Theoretical results are illustrated on numerical examples.

We complete the first step towards the resolution of several decades-old challenges in disturbance-robust adaptive control. For a scalar linear system with an unknown parameter for which no a priori bound is given, with a disturbance that is of unlimited magnitude and possibly persistent (not square integrable), and without a persistency of excitation necessarily verified by the state, we consider the problems of (practical) gain assignment relative to the disturbance. We provide a solution to these heretofore unsolved feedback design problems with the aid of infinite-dimensional nonlinear feedback employing distributed delay of the state and input itself. Specifically, in addition to (0) the global boundedness of the infinite-dimensional state of the closed-loop system when the disturbance is present, we establish (1) practical input-to-output stability with assignable asymptotic gain from the disturbance to the plant state; (2) assignable exponential convergence rate; and (3) assignable radius of the residual set. The accompanying identifier in our adaptive controller guarantees (4) boundedness of the parameter estimate even when disturbances are present; (5) an ultimate estimation error which is proportional to the magnitude of the disturbance with assignable gain when there exists sufficient excitation of the state; and (6) exact parameter estimation in finite-time when the disturbance is absent and there is sufficient excitation. Among our results, one reveals a tradeoff between "learning capacity" and "disturbance robustness:" the less sensitive the identifier is to the disturbance, the less likely it is to learn the parameter., Comment: 28 pages, submitted for possible publication to SIAM Journal on Control and Optimization

In this paper, we investigate the problem of source seeking with a unicycle in the presence of local extrema. Our study is motivated by the fact that most of the existing source seeking methods follow the gradient direction of the signal function and thus only lead to local convergence into a neighborhood of the nearest local extremum. So far, only a few studies present ideas on how to overcome local extrema in order to reach a global extremum. None of them apply to second-order (force- and torque-actuated) nonholonomic vehicles. We consider what is possibly the simplest conceivable algorithm for such vehicles, which employs a constant torque and a translational/surge force in proportion to an approximately differentiated measured signal. We show that the algorithm steers the unicycle through local extrema towards a global extremum. In contrast to the previous extremum-seeking studies, in our analysis we do not approximate the gradient of the objective function but of the objective function's local spatial average. Such a spatially averaged objective function is expected to have fewer critical points than the original objective function. Under suitable assumptions on the averaged objective function and on sufficiently strong translational damping, we show that the control law achieves practical uniform asymptotic stability and robustness to sufficiently weak measurement noise and disturbances to the force and torque inputs.

Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles functional coefficients, dependent on the spatial variables, and gain functions dependent on space as well. The designs of gains for controllers and observers for PDEs, such as PDE backstepping, are mappings of system model functions into gain functions. These infinite dimensional nonlinear operators are given in an implicit form through PDEs, in spatial variables, which need to be solved to determine the gain function for each new functional coefficient of the PDE. The need for solving such PDEs can be eliminated by learning and approximating the said design mapping in the form of a neural operator. Learning the neural operator requires a sufficient number of prior solutions for the design PDEs, offline, as well as the training of the operator. In recent work, we developed the neural operators for PDE backstepping designs for first order hyperbolic PDEs. Here we extend this framework to the more complex class of parabolic PDEs. The key theoretical question is whether the controllers are still stabilizing, and whether the observers are still convergent, if they employ the approximate functional gains generated by the neural operator. We provide affirmative answers to these questions, namely, we prove stability in closed loop under gains produced by neural operators. We illustrate the theoretical results with numerical tests and publish our code on github. The neural operators are three orders of magnitude faster in generating gain functions than PDE solvers for such gain functions. This opens up the opportunity for the use of this neural operator methodology in adaptive control and in gain scheduling control for nonlinear PDEs., Comment: 17 pages, 8 figures

We introduce a framework for eliminating the computation of controller gain functions in PDE control. We learn the nonlinear operator from the plant parameters to the control gains with a (deep) neural network. We provide closed-loop stability guarantees (global exponential) under an NN-approximation of the feedback gains. While, in the existing PDE backstepping, finding the gain kernel requires (one offline) solution to an integral equation, the neural operator (NO) approach we propose learns the mapping from the functional coefficients of the plant PDE to the kernel function by employing a sufficiently high number of offline numerical solutions to the kernel integral equation, for a large enough number of the PDE model's different functional coefficients. We prove the existence of a DeepONet approximation, with arbitrarily high accuracy, of the exact nonlinear continuous operator mapping PDE coefficient functions into gain functions. Once proven to exist, learning of the NO is standard, completed "once and for all" (never online) and the kernel integral equation doesn't need to be solved ever again, for any new functional coefficient not exceeding the magnitude of the functional coefficients used for training. We also present an extension from approximating the gain kernel operator to approximating the full feedback law mapping, from plant parameter functions and state measurement functions to the control input, with semiglobal practical stability guarantees. Simulation illustrations are provided and code is available on github. This framework, eliminating real-time recomputation of gains, has the potential to be game changing for adaptive control of PDEs and gain scheduling control of nonlinear PDEs. The paper requires no prior background in machine learning or neural networks., Comment: 16 Pages, 8 Figures