arXiv subjects

Tiago Roux Oliveira

Publications and source records attributed to Tiago Roux Oliveira.

At least 37 records · Page 2Linked to original sources

arXiv2025

Unit-Vector Control Design under Saturating Actuators

This paper deals with unit vector control design for multivariable polytopic uncertain systems under saturating actuators. For that purpose, we propose LMI-based conditions to design the unit vector control gain such that the origin of the closed-loop system is finite-time stable. Moreover, an optimization problem is provided to obtain an enlarged estimate of the region of attraction of the equilibrium point for the closed-loop system, where the convergence of trajectories is ensured even in the presence of saturation functions. Numerical simulations illustrate the effectiveness of the proposed approach.

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arXiv2025

Multivariable Extremum Seeking Unit-Vector Control Design

This paper investigates multivariable extremum seeking using unit-vector control. By employing the gradient algorithm and a polytopic embedding of the unknown Hessian matrix, we establish sufficient conditions, expressed as linear matrix inequalities, for designing the unit-vector control gain that ensures finite-time stability of the origin of the average closed-loop error system. Notably, these conditions enable the design of non-diagonal control gains, which provide extra degrees of freedom to the solution. The convergence of the actual closed-loop system to a neighborhood of the unknown extremum point is rigorously proven through averaging analysis for systems with discontinuous right-hand sides. Numerical simulations illustrate the efficacy of the proposed extremum seeking control algorithm.

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arXiv2025

Extremum Seeking Control for Multivariable Maps under Actuator Saturation

This paper deals with the gradient-based extremum seeking control for multivariable maps under actuator saturation. By exploiting a polytopic embedding of the unknown Hessian, we derive a LMI-based synthesis condition to ensure that the origin of the average closed-loop error system is exponentially stable. Then, the convergence of the extremum seeking control system under actuator saturation to the unknown optimal point is proved by employing Lyapunov stability and averaging theories. Numerical simulations illustrate the efficacy of the proposed approach.

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arXiv2025

Multivariable Stochastic Newton-Based Extremum Seeking with Delays

This paper presents a Newton-based stochastic extremum-seeking control method for real-time optimization in multi-input systems with distinct input delays. It combines predictor-based feedback and Hessian inverse estimation via stochastic perturbations to enable delay compensation with user-defined convergence rates. The method ensures exponential stability and convergence near the unknown extremum, even under long delays. It extends to multi-input, single-output systems with cross-coupled channels. Stability is analyzed using backstepping and infinite-dimensional averaging. Numerical simulations demonstrate its effectiveness in handling time-delayed channels, showcasing both the challenges and benefits of real-time optimization in distributed parameter settings.

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arXiv2025

Event-Triggered Newton-Based Extremum Seeking Control

This paper proposes the incorporation of static event-triggered control in the actuation path of Newton-based extremum seeking and its comparison with the earlier gradient version. As in the continuous methods, the convergence rate of the gradient approach depends on the unknown Hessian of the nonlinear map to be optimized, whereas the proposed event-triggered Newton-based extremum seeking eliminates this dependence, becoming user-assignable. This is achieved by means of a dynamic estimator for the Hessian's inverse, implemented as a Riccati equation filter. Lyapunov stability and averaging theory for discontinuous systems are applied to analyze the closed-loop system. Local exponential practical stability is guaranteed to a small neighborhood of the extremum point of scalar and static maps. Numerical simulations illustrate the advantages of the proposed approach over the previous gradient method, including improved convergence speed, followed by a reduction in the amplitude and updating frequency of the control signals.

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arXiv2025

Lie-Bracket Nash Equilibrium Seeking with Bounded Update Rates for Noncooperative Games

This paper proposes a novel approach for local convergence to Nash equilibrium in quadratic noncooperative games based on a distributed Lie-bracket extremum seeking control scheme. This is the first instance of noncooperative games being tackled in a model-free fashion integrated with the extremum seeking method of bounded update rates. In particular, the stability analysis is carried out using Lie-bracket approximation and Lyapunov's direct method. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an example in an oligopoly setting.

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arXiv2024

Slow and fast dynamics in measure functional differential equations with state-dependent delays through averaging principles and applications to extremum seeking

This paper investigates a new class of equations called measure functional differential equations with state-dependent delays. We establish the existence and uniqueness of solutions and present a discussion concerning the appropriate phase space to define these equations. Also, we prove a version of periodic averaging principle to these equations. This type of result was completely open in the literature. These equations involving measure bring the advantage to encompass others such as impulsive, dynamic equations on time scales and difference equations, expanding their application potential. Additionally, we apply our theoretical insights to a real-time optimization strategy, using extremum seeking to validate the stability of an innovative algorithm under state-dependent delays. This application confirm the relevance of our findings in practical scenarios, offering valuable tools for advanced control system design. Our research provides significant contributions to the mathematical field and suggests new directions for future technological developments.

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arXiv2024

A Systematic LMI Approach to Design Multivariable Sliding Mode Controllers

This paper deals with sliding mode control for multivariable polytopic uncertain systems. We provide systematic procedures to design variable structure controllers (VSCs) and unit-vector controllers (UVCs). Based on suitable representations for the closed-loop system, we derive sufficient conditions in the form of linear matrix inequalities (LMIs) to design the robust sliding mode controllers such that the origin of the closed-loop system is globally stable in finite time. Moreover, by noticing that the reaching time depends on the initial condition and the decay rate, we provide convex optimization problems to design robust controllers by considering the minimization of the reaching time associated with a given set of initial conditions. Two examples illustrate the effectiveness of the proposed approaches.

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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

Gradient-Based Stochastic Extremum-Seeking Control for Multivariable Systems with Distinct Input Delays

This paper addresses the design and analysis of a multivariable gradient-based stochastic extremum-seeking control method for multi-input systems with arbitrary input delays. The approach accommodates systems with distinct time delays across input channels and achieves local exponential stability of the closed-loop system, guaranteeing convergence to a small neighborhood around the extremum point. By incorporating phase compensation for dither signals and a novel predictor-feedback mechanism with averaging-based estimates of the unknown gradient and Hessian, the proposed method overcomes traditional challenges associated with arbitrary, distinct input delays. Unlike previous work on deterministic multiparameter extremum-seeking with distinct input delays, this stability analysis is achieved without using backstepping transformations, simplifying the predictor design and enabling a more straightforward implementation. Specifically, the direct application of Artstein's reduction approach results in delay- and system-dimension-independent convergence rates, enhancing practical applicability. A numerical example illustrates the robust performance and advantages of the proposed delay-compensated stochastic extremum-seeking method.

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arXiv2024

Population Control of Giardia lamblia

Giardia lamblia is a flagellate intestinal protozoan with global distribution causing the disease known as giardiasis. This parasite is responsable for 35.1% of outbreaks of diarrhea caused by contaminated water which and mainly affects children in whom it can cause physical and cognitive impairment. In this paper, we consider a model of population dynamics to represent the behavior of Giardia lamblia in vitro, taking into account its mutation characteristic that guarantees to the protozoan resistance to the drug metronidazole. Different from what is found in the literature, it is pursued as the control objective the extermination of the protozoan considering that the parameters of the model are uncertain and only the partial measurement of the state vector is possible. On these assumptions, a control law is designed and the stability of the closed-loop system is rigorously proved. Simulation and experimental results illustrate the benefits of the proposed population control method of Giardia lamblia.

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arXiv2024

Multivariable Extremum Seeking Control for Dynamic Maps through Sliding Modes and Periodic Switching Function

This paper presents the design of an extremum seeking controller based on sliding modes and cyclic search for real-time optimization of non-linear multivariable dynamic systems. These systems have arbitrary relative degree, compensated by the technique of time-scaling. The resulting approach guarantees global convergence of the system output to a small neighborhood of the optimum point. To corroborate with the theoretical results, numerical simulations are presented considering a system with two inputs and one output, which rapidly converges to the optimal parameters of the objective function.

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arXiv2024

Extremum Seeking Control for Scalar Maps with Distributed Diffusion PDEs

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.

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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

Global Output-Feedback Extremum Seeking Control with Source Seeking Experiments

This paper discusses the design of an extremum seeking controller that relies on a monitoring function for a class of SISO uncertain nonlinear systems characterized by arbitrary and uncertain relative degree. Our demonstration illustrates the feasibility of achieving an arbitrarily small proximity to the desired optimal point through output feedback. The core concept involves integrating a monitoring function with a norm state observer for the unitary relative degree case and its expansion to arbitrary relative degrees by means of the employment of a time-scaling technique. Significantly, our proposed scheme attains the extremum of an unknown nonlinear mapping across the entire domain of initial conditions, ensuring global convergence and stability for the real-time optimization algorithm. Furthermore, we provide tuning rules to ensure convergence to the global maximum in the presence of local extrema. To validate the effectiveness of the proposed approach, we present a numerical example and apply it to a source-seeking problem involving a cart-track linear positioning servomechanism. Notably, the cart lacks the ability to sense its velocity or the source's position, but can detect the source of a light signal of unknown concentration field.

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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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arXiv2024

Second-Order Newton-Based Extremum Seeking for Multivariable Static Maps

A second-order Newton-based extremum seeking (SONES) algorithm is presented to estimate directional inflection points for multivariable static maps. The design extends the first-order Newton-based extremum seeking algorithm that drives the system toward its peak point. This work provides perturbation matrices to estimate the second- and third-order derivatives necessary for implementation of the SONES. A set of conditions are provided for the probing frequencies that ensure accurate estimation of the derivatives. A differential Riccati filter is used to calculate the inverse of the third-order derivative. The local stability of the new algorithm is proven for general multivariable static maps using averaging analysis. The proposed algorithm ensures uniform convergence toward directional inflection point without requiring information about the curvature of the map and its gradient. Simulation results show the effectiveness of the proposed algorithm.

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arXiv2023

Event-Triggered Extremum Seeking Control Systems

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.

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