arXiv subjects

Emilia Fridman

Publications and source records attributed to Emilia Fridman.

At least 19 recordsLinked to original sources

arXiv2026

Fixed-time stabilization of systems with nilpotent matrices in the presence of large unknown delays

In this paper, we study state-feedback fixed-time stabilization of continuous-time perturbed linear systems with nilpotent matrices in the presence of unknown large constant input/measurement delays with known bounds. By using an additional artificial delay, we design a feedback that for a precisely known linear system with a nilpotent matrix drives the state to zero in fixed time (which is larger than the bound of the unknown delay) and achieves input-to-state stability with a large exponential decay rate in the presence of additive disturbances and small system matrix perturbations and nonlinearities. For perturbed single integrators, fixed-time stabilization is achieved in the case of fast-varying unknown measurement delays with known bounds. The results are extended to discrete-time systems. Numerical simulations illustrate the efficiency of the results.

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arXiv2026

Output Regulation for Linear Parabolic Systems Using Finite-Dimensional Tracking-Error-Based Control

This paper addresses the output regulation problem for 1-D diffusion-reaction system, where both the disturbance and reference signals are generated by an unstable exosystem. We propose a constructive approach to the design of a finite-dimensional tracking error-based regulator for this class of possibly unstable systems. Based on the regulator equations, a combined plant is derived, converting the output regulation problem into a partial stabilization problem for the combined system. Using the modal decomposition method, the observability of the truncated modes is characterized under appropriate transmission zeros and observability conditions. For the controller design, unlike in the standard stabilization case, where the observer gain is dependent on the unstable modes only, the observer gain here has full order due to the coupling introduced by the exosystem. We prove that the observer gain can be designed such that its norm remains uniformly bounded with respect to the dimension of the observer. LMI-based conditions are provided for determining the observer dimension, and it is shown that the LMI is feasible for a sufficiently large dimension. Finally, the output regulation problem in the presence of unknown time-varying measurement delays is analyzed. Numerical examples are provided to validate the theoretical results.

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arXiv2026

Extremum Seeking of Static Maps in the Presence of Unknown Large Time-Varying Delays

In this paper, we present the discrete-time unbiased extremum seeking (ES) algorithm for n-dimensional (nD) static quadratic maps in the presence of unknown time-varying measurement delays bounded by known constants which can be large. The existing ES results in the presence of large delays are usually confined to known constant or slowly-varying delays, which is restrictive. We provide the first ES algorithm, which is robust with respect to unknown large time-varying delays. Moreover, we achieve the unbiased exponential convergence. We manage with such delays by choosing dithers with frequencies of the order of \sqrt{\epsilon}, where the small parameter {\epsilon} > 0 appears in the dynamics of the real-time estimator. As expected, larger delays lead to a slower convergence. We provide qualitative and quantitative results based on the averaging analysis via delay-free transformation. For the quantitative bounds on the controller parameters that ensure the exponential unbiased convergence of the ES system, we assume that the Hessian of the map is uncertain and lies within a known range. Differently from its continuous-time counterpart, the small parameter in the discrete-time case defines the decay rate of the estimation error system, making a quantitative bound on this parameter particularly important. We present also constructive conditions for the practical stability of the classical ES system. Our results are semi-global for globally quadratic maps, while for locally quadratic static maps, we provide a bound on the region of convergence. Our analysis shows that appropriate ES parameters can be found for any large unknown time-varying bounded delay. A numerical example highlights the efficiency of the method.

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arXiv2025

Constructive boundary observer-based control of high-dimensional semilinear heat equations

This paper presents a constructive finite-dimensional output-feedback design for semilinear $M$-dimensional ($M\geq 2$) heat equations with boundary actuation and sensing. A key challenge in high dimensions is the slower growth rate of the Laplacian eigenvalues. The novel features of our modal-decomposition-based design, which allows to enlarge Lipschitz constants, include a larger class of shape functions that may be distributed over a part of the boundary only, the corresponding lifting transformation and the full-order controller gain found from the design LMIs. We further analyze the robustness of the closed-loop system with respect to either multiplicative noise (vanishing at the origin) or additive noise (persistent). Effective LMI conditions are provided for specifying the minimal observer dimension and maximal Lipschitz constants that preserve the stability (mean-square exponential stability for multiplicative noise and noise-to-state stability for additive noise). Numerical examples for 2D and 3D cases demonstrate the efficacy and advantages of our method.

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arXiv2025

Distributed Time-Varying Optimization via Unbiased Extremum Seeking

This paper proposes a novel distributed optimization framework that addresses time-varying optimization problems without requiring explicit derivative information of the objective functions. Traditional distributed methods often rely on derivative computations, limiting their applicability when only real-time objective function measurements are available. Leveraging unbiased extremum seeking, we develop continuous-time algorithms that utilize local measurements and neighbor-shared data to collaboratively track time-varying optima. Key advancements include compatibility with directed communication graphs, customizable convergence rates (asymptotic, exponential, or prescribed-time), and the ability to handle dynamically evolving objectives. By integrating chirpy probing signals with time-varying frequencies, our unified framework achieves accelerated convergence while maintaining stability under mild assumptions. Theoretical guarantees are established through Lie bracket averaging and Lyapunov-based analysis, with linear matrix inequality conditions ensuring rigorous convergence. Numerical simulations validate the effectiveness of the algorithms.

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arXiv2025

Output-Feedback Control of the Semilinear Heat Equation via the $L^2$ Residue Separation and Harmonic Inequality

A popular approach to designing finite-dimensional boundary controllers for partial differential equations (PDEs) is to decompose the PDE into independent modes and focus on the dominant ones while neglecting highly damped residual modes. However, the neglected modes can adversely affect the overall system performance, causing spillover. The $L^2$ residue separation method was recently introduced to eliminate spillover in the state-feedback control design. In this paper, we extend this method to finite-dimensional output-feedback control, where the output is contaminated by the residual modes. To deal with the output residue, we introduce a new harmonic inequality that optimally bounds it. We develop the approach for a 1D heat equation with unknown nonlinearity, where boundary temperature measurements are used to control heat flux at the opposite boundary. By exploiting the connection between $L^2$ residue separation and $H_\infty$ theory, we show that the class of admissible nonlinearities can only increase with higher controller order.

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arXiv2025

Improved residual mode separation for finite-dimensional control of PDEs: application to the Euler-Bernoulli beam

We consider a simply-supported Euler-Bernoulli beam with viscous and Kelvin--Voigt damping. Our objective is to attenuate the effect of an unknown distributed disturbance using one piezoelectric actuator. We show how to design a suitable $H_\infty$ state-feedback controller based on a finite number of dominating modes. If the remaining (infinitely many) modes are ignored, the calculated $L^2$ gain is wrong. This happens because of the spillover phenomenon that occurs when the effect of the control on truncated modes is not accounted for in the feedback design. We propose a simple modification of the $H_\infty$ cost that prevents spillover. The key idea is to treat the control as a disturbance in the truncated modes and find the corresponding $L^2$ gains using the bounded real lemma. These $L^2$ gains are added to the control weight in the $H_\infty$ cost for the dominating modes, which prevents spillover. A numerical simulation of an aluminum beam with realistic parameters demonstrates the effectiveness of the proposed method. The presented approach is applicable to other types of PDEs, such as the heat, wave, and Kuramoto-Sivashinsky equations, as well as their semilinear versions. While this work focuses on $H_\infty$ control, the same methodology can be applied to guaranteed cost control, regional stability analysis, input-to-state stability, and systems with time-varying delays, including sampled-data systems.

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arXiv2025

Extremum Seeking for Linear Time-Varying Systems with Unknown Control Directions

We consider bounded extremum seeking controls for time-varying linear systems with uncertain coefficient matrices and measurement uncertainty. Using a new change of variables, Lyapunov functions, and a comparison principle, we provide practical exponential stability bounds for the states of the closed loop systems that hold for all nonnegative times. For the first time for linear time-varying systems with unknown control directions, we consider bounded extremum seeking controls in the presence of uncertain time-varying input delays with small time-varying delay uncertainties, and we provide reduction model controllers to compensate for the constant part of the delays.

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arXiv2024

Privacy and Security Trade-off in Interconnected Systems with Known or Unknown Privacy Noise Covariance

This paper is concerned with the security problem for interconnected systems, where each subsystem is required to detect local attacks using locally available information and the information received from its neighboring subsystems. Moreover, we consider that there exists an additional eavesdropper being able to infer the private information by eavesdropping transmitted data between subsystems. Then, a privacy-preserving method is employed by adding privacy noise to transmitted data, and the privacy level is measured by mutual information. Nevertheless, adding privacy noise to transmitted data may affect the detection performance metrics such as detection probability and false alarm probability. Thus, we theoretically analyze the trade-off between the privacy and the detection performance. An optimization problem with maximizing both the degree of privacy preservation and the detection probability is established to obtain the covariance of the privacy noise. In addition, the attack detector of each subsystem may not obtain all information about the privacy noise. We further theoretically analyze the trade-off between the privacy and the false alarm probability when the attack detector has no knowledge of the privacy noise covariance. An optimization problem with maximizing the degree of privacy preservation with guaranteeing a bound of false alarm distortion level is established to obtain {\color{black}{the covariance of the privacy noise}}. Moreover, to analyze the effect of the privacy noise on the detection probability, we consider that each subsystem can estimate the unknown privacy noise covariance by the secondary data. Based on the estimated covariance, we construct another attack detector and analyze how the privacy noise affects its detection performance. Finally, a numerical example is provided to verify the effectiveness of theoretical results.

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arXiv2024

Delayed finite-dimensional observer-based control of 2D linear parabolic PDEs

Recently, a constructive method was suggested for finite-dimensional observer-based control of 1D linear heat equation, which is robust to input/output delays. In this paper, we aim to extend this method to the 2D case with general time-varying input/output delays (known output delay and unknown input delay) or sawtooth delays (that correspond to network-based control). We use the modal decomposition approach and consider boundary or non-local sensing together with non-local actuation, or Neumann actuation with non-local sensing. To compensate the output delay that appears in the infinite-dimensional part of the closed-loop system, for the first time for delayed PDEs we suggest a vector Lyapunov functional combined with the recently introduced vector Halanay inequality. We provide linear matrix inequality (LMI) conditions for finding the observer dimension and upper bounds on delays that preserve the exponential stability. We prove that the LMIs are always feasible for large enough observer dimension and small enough upper bounds on delays. A numerical example demonstrates the efficiency of our method and shows that the employment of vector Halanay's inequality allows for larger delays than the classical scalar Halanay inequality for comparatively large observer dimension.

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arXiv2023

Stabilization of underactuated linear coupled reaction-diffusion PDEs via distributed or boundary actuation

This work concerns the exponential stabilization of underactuated linear homogeneous systems of m parabolic partial differential equations (PDEs) in cascade (reaction-diffusion systems), where only the first state is controlled either internally or from the right boundary and in which the diffusion coefficients are distinct. For the distributed control case, a proportional-type stabilizing control is given explicitly. After applying modal decomposition, the stabilizing law is based on a transformation for the ordinary differential equations (ODE) system corresponding to the comparatively unstable modes into a target one, where the calculation of the stabilization law is independent of the arbitrarily large number of these modes. This is achieved by solving generalized Sylvester equations recursively. For the boundary control case, under appropriate sufficient conditions on the coupling matrix (reaction term), the proposed controller is dynamic. A dynamic extension technique via trigonometric change of variables that places the control internally is first performed. Then, modal decomposition is applied followed by a state transformation of the ODE system, which must be stabilized in order to be written in a form where a dynamic law can be established. For both distributed and boundary control systems, a constructive and scalable stabilization algorithm is proposed, as the choice of the controller gains is independent of the number of unstable modes and only relies on the stabilization of the reaction term. The present approach solves the problem of stabilization of underactuated systems when in the presence of distinct diffusion coefficients, the problem is not directly solvable, similarly to the scalar PDE case. Keywords: Linear parabolic PDE systems, underactuated systems, stabilization, modal decomposition

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arXiv2023

Extremum seeking in the presence of large delays via time-delay approach to averaging

In this paper, we study gradient-based classical extremum seeking (ES) for uncertain n-dimensional (nD) static quadratic maps in the presence of known large constant distinct input delays and large output constant delay with a small time-varying uncertainty. This uncertainty may appear due to network-based measurements. We present a quantitative analysis via a time-delay approach to averaging. We assume that the Hessian has a nominal known part and norm-bounded uncertainty, the extremum point belongs to a known box, whereas the extremum value to a known interval. By using the orthogonal transformation, we first transform the original static quadratic map into a new one with the Hessian containing a nominal diagonal part. We apply further a time-delay transformation to the resulting ES system and arrive at a time-delay system, which is a perturbation of a linear time-delay system with constant coefficients. Given large delays, we choose appropriate gains to guarantee stability of this linear system. To find a lower bound on the dither frequency for practical stability, we employ variation of constants formula and exploit the delay-dependent positivity of the fundamental solutions of the linear system with their tight exponential bounds. Sampled-data ES in the presence of large distinct input delays is also presented. Explicit conditions in terms of simple scalar inequalities depending on tuning parameters and delay bounds are established to guarantee the practical stability of the ES control systems. We show that given any large delays and initial box, by choosing appropriate gains we can achieve practical stability for fast enough dithers and small enough uncertainties.

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arXiv2023

Internal stabilization of three interconnected semilinear reaction-diffusion PDEs with one actuated state

This work deals with the exponential stabilization of a system of three semilinear parabolic partial differential equations (PDEs), written in a strict feedforward form. The diffusion coefficients are considered distinct and the PDEs are interconnected via both a reaction matrix and a nonlinearity. Only one of the PDEs is assumed to be controlled internally, thereby leading to an underactuated system. Constructive and efficient control of such underactuated systems is a nontrivial open problem, which has been solved recently for the linear case. In this work, these results are extended to the semilinear case, which is highly challenging due the interconnection that is introduced by the nonlinearity. Modal decomposition is employed, where due to nonlinearity, the finite-dimensional part of the solution is coupled with the infinite-dimensional tail. A transformation is then performed to map the finite-dimensional part into a target system, which allows for an efficient design of a static linear proportional state-feedback controller. Furthermore, a high-gain approach is employed in order to compensate for the nonlilinear terms. Lyapunov stability analysis is performed, leading to LMI conditions guaranteeing exponential stability with arbitrary decay rate. The LMIs are shown to always be feasible, provided the number of actuators and the value of the high gain parameter are large enough. Numerical examples show the efficiency of the proposed approach.

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arXiv2023

Detectability and global observer design for 2D Navier-Stokes equations with uncertain inputs

We present simulation friendly detectability conditions for 2D Navier-Stokes Equation (NSE) with periodic boundary conditions, and describe a generic class of ``detectable'' observation operators: it includes pointwise evaluation of NSE's solution at interpolation nodes, and spatial average measurements. For ``detectable'' observation operators we design a global infinite-dimensional observer for NSE with uncertain possibly destabilizing inputs: in our numerical experiments we illustrate $H^1$-sensitivity of NSE to small perturbations of initial conditions, yet the observer converges for known and uncertain inputs.

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arXiv2023

A Robust Time-Delay Approach to Extremum Seeking via ISS Analysis of the Averaged System

For N-dimensional (ND) static quadratic map, we present a time-delay approach to gradient-based extremum seeking (ES) both, in the continuous and, for the first time, the discrete domains. As in the recently introduced (for 2D maps in the continuous domain), we transform the system to the time-delay one (neutral type system in the form of Hale in the continuous case). This system is O($\varepsilon$)-perturbation of the averaged linear ODE system, where $\varepsilon$ is a period of averaging. We further explicitly present the neutral system as the linear ODE, where O($\varepsilon$)-terms are considered as disturbances with distributed delays of the length of the small parameter $\varepsilon$. Regional input-to-state stability (ISS) analysis is provided by employing a variation of constants formula that greatly simplifies the previously used analysis via Lyapunov-Krasovskii (L-K) method, simplifies the conditions and improves the results. Examples from the literature illustrate the efficiency of the new approach, allowing essentially large uncertainty of the Hessian matrix with bounds on $\varepsilon$ that are not too small.

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arXiv2022

Internal stabilization of an underactuated linear parabolic system via modal decomposition (extended version)

This work concerns the internal stabilization of underactuated linear systems of $m$ heat equations in cascade, where the control is placed internally in the first equation only and the diffusion coefficients are distinct. Combining the modal decomposition method with a recently introduced state-transformation approach for observation problems, a proportional-type stabilizing control is given explicitly. It is based on a transformation for the ODE system corresponding to the comparatively unstable modes into a target one, where calculation of the stabilization law is independent of the arbitrarily large number of them and it is achieved by solving generalized Sylvester equations recursively. This provides a finite-dimensional counterpart of a recently introduced infinite-dimensional one, which led to Lyapunov stabilization. The present approach answers to the problem of stabilization with actuators not appearing in all the states and when boundary control results do not apply.

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arXiv2022

Delayed finite-dimensional observer-based control of 1D heat equation under Neumann actuation

Recently a constructive method was introduced for finite-dimensional observer-based control of 1D parabolic PDEs. In this paper we present an improved method in terms of the reduced-order LMIs (that significantly shorten the computation time) and introduce predictors to manage with larger delays. We treat the case of a 1D heat equation under Neumann actuation and non-local measurement, that has not been studied yet. We apply modal decomposition and prove $L^2$ exponential stability by a direct Lyapunov method. We provide reduced-order LMI conditions for finding the observer dimension $N$ and resulting decay rate. The LMI dimension does not grow with $N$. The LMI is always feasible for large $N$, and feasibility for $N$ implies feasibility for $N+1$. For the first time we manage with delayed implementation of the controller in the presence of fast-varying (without any constraints on the delay-derivative) input and output delays. To manage with larger delays, we construct classical observer-based predictors. For the known input delay, the LMIs dimension does not grow with $N$, whereas for unknown one the LMIs dimension grows, but it is ssentially smaller than in the existing results. A numerical example demonstrates the efficiency of our method.

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arXiv2021

Global finite-dimensional observer-based stabilization of a semilinear heat equation with large input delay

We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lypunov $H^1$-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a constant input delay $r>0$. To compensate the delay, we introduce a chain of $M$ sub-predictors that leads to a nonlinear closed-loop ODE system, coupled with nonlinear infinite-dimensional tail ODEs. We provide LMIs for $H^1$-stability and prove that for any $r>0$, the LMIs are feasible provided $M$ and $N$ are large enough and the Lipschitz constant is small enough. Numerical examples demonstrate the efficiency of the proposed approach.

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