Building on the recent work by Geshkovski et al. (2023) which provides an interacting particle system interpretation of Transformers with a continuous-time evolution, we study the controllability attributes of the cor...
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Building on the recent work by Geshkovski et al. (2023) which provides an interacting particle system interpretation of Transformers with a continuous-time evolution, we study the controllability attributes of the corresponding continuity equation across the landscape of probability space curves. In particular, we consider the parameters of the Transformer's continuous-time evolution as control inputs. We prove that given an absolutely continuous probability measure and a non-local Lipschitz velocity field that satisfy a continuity equation, there exist control inputs such that the measure and the non-local velocity field of the Transformer's continuous-time evolution approximate them, respectively, in the p-Wasserstein and L-p-sense, where 1 <= p < infinity.
We consider a 1D semilinear reaction-diffusion system with controlled heat flux at one of the boundaries. We design a finite-dimensional state-feedback controller guaranteeing that a given quadratic cost does not exce...
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We consider a 1D semilinear reaction-diffusion system with controlled heat flux at one of the boundaries. We design a finite-dimensional state-feedback controller guaranteeing that a given quadratic cost does not exceed a prescribed value for all nonlinearities with a predefined Lipschitz constant. To this end, we perform modal decomposition and truncate the highly damped (residue) modes. To deal with the nonlinearity that couples the residue and dominating modes, we combine the direct Lyapunov approach with the S-procedure and Parseval's identity. The truncation may lead to spillover: the ignored modes can deteriorate the overall system performance. Our main contribution is spillover avoidance via the L-2 separation of the residue. Namely, we calculate the L-2 input-to-state gains for the residue modes and add them to the control weight in the quadratic cost used to design a controller for the dominating modes. A numerical example demonstrates that the proposed idea drastically improves both the admissible Lipschitz constants and guaranteed cost bound compared to the recently introduced direct Lyapunov method.
This letter presents the design and analysis of gradient extremum seeking applied to scalar static maps within the context of infinite-dimensional dynamics governed by Partial Differential Equations (PDEs) of wave typ...
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This letter presents the design and analysis of gradient extremum seeking applied to scalar static maps within the context of infinite-dimensional dynamics governed by Partial Differential Equations (PDEs) of wave type featuring a small amount of Kelvin-Voigt damping. Notably, this particular class of PDEs for extremum seeking still needs to be explored in the existing literature. First, to compensate for the influence of PDE actuation dynamics, we employ a boundary control law via backstepping transformation and averaging-based estimates of the gradient and Hessian. Finally, we prove the local exponential convergence to a small neighborhood surrounding the unknown optimal point by means of an Input-to-State Stability analysis, as well as by employing Lyapunov functionals and averaging theory in infinite dimensions.
In this paper, we provide a constructive method of specifying initial/boundary conditions for linear unforced systems of partial differential equations (PDEs) with real constant coefficients. In the literature, one of...
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In this paper, we provide a constructive method of specifying initial/boundary conditions for linear unforced systems of partial differential equations (PDEs) with real constant coefficients. In the literature, one of the ways of specifying initial/boundary data is by specifying the values of certain derivatives of the solution trajectories at the origin. These derivatives correspond to a standard monomial set and is obtained using Grobner basis. This method of specifying initial data often lacks physical interpretation. In this paper, we propose a method of specifying initial/boundary data by restricting trajectories to subsets of the domain having an algebraic structure given by a hyper-cubic strip of infinite length and finite width;such special subsets are known as characteristic sets. We provide a method of constructing characteristic sets of this form. We further show that every unforced system of linear PDEs with real constant coefficients admits a characteristic set of the above-mentioned form. This provides a complete answer to the question of initial/boundary conditions for systems of linear PDEs with real constant coefficients.
An increasing speed of a motion change in conjunction with lightweight elastic structures lead to large elastic deflections. In order to maintain a sufficient positioning accuracy, vibration damping is mandatory. This...
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An increasing speed of a motion change in conjunction with lightweight elastic structures lead to large elastic deflections. In order to maintain a sufficient positioning accuracy, vibration damping is mandatory. This can be achieved directly by a feedback loop or indirectly by shaping the reference signal. Based on a G2-norm optimal filter that takes the desired rigid body acceleration and elastic mode dynamics into account and returns a force signal, a reshaped velocity signal for reducing the elastic oscillation is generated. The method is validated on a test setup comprising a linearly moving slide with an upright beam and measurements show a major reduction in the beam oscillation.
In this technical note, we consider the stability properties of a viscously damped Timoshenko beam equations with spatially varying parameters. With the help of the port-Hamiltonian framework, we first prove the exist...
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In this technical note, we consider the stability properties of a viscously damped Timoshenko beam equations with spatially varying parameters. With the help of the port-Hamiltonian framework, we first prove the existence of solutions and show, by the use of an appropriate Lyapunov function, that the system is exponentially stable and has an explicit decay rate. The explicit exponential bound is computed for an illustrative example for which we provide some numerical simulations.
This article presents an online parameter identification scheme for advection-diffusion processes using data collected by a mobile sensor network. The advection-diffusion equation is incorporated into the information ...
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This article presents an online parameter identification scheme for advection-diffusion processes using data collected by a mobile sensor network. The advection-diffusion equation is incorporated into the information dynamics associated with the trajectories of the mobile sensors. A constrained cooperative Kalman filter is developed to provide estimates of the field values and gradients along the trajectories of the mobile sensors so that the temporal variations in the field values can be estimated. This leads to a co-design scheme for state estimation and parameter identification for advection-diffusion processes that is different from comparable schemes using sensors installed at fixed spatial locations. Using state estimates from the constrained cooperative Kalman filter, a recursive least-square (RLS) algorithm is designed to estimate unknown model parameters of the advection-diffusion processes. Theoretical justifications are provided for the convergence of the proposed cooperative Kalman filter by deriving a set of sufficient conditions regarding the formation shape and the motion of the mobile sensor network. Simulation and experimental results show satisfactory performance and demonstrate the robustness of the algorithm under realistic uncertainties and disturbances.
The high-voltage direct current (HVDC) transmission based on modular multilevel converter shows the weak feed characteristic of limited fault current amplitude when connected to the traditional AC power grid, which le...
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The high-voltage direct current (HVDC) transmission based on modular multilevel converter shows the weak feed characteristic of limited fault current amplitude when connected to the traditional AC power grid, which leads to the risk of sensitivity reduction or even failure of the traditional pilot protection. Here, a modular multi-level converter (MMC)-HVDC AC side transmission line protection scheme based on positive sequence fault component current ratio is proposed. The positive sequence fault component current of the MMC-HVDC side is calculated by using the positive sequence fault component current of the power grid side, and the fault is identified through the significant difference between the ratio of the positive sequence fault component current of the MMC-HVDC side and the actual positive sequence fault component current of the MMC-HVDC side in the case of internal and external faults. This method has the advantages of simple principle, low requirement for synchronous data, not affected by capacitive current and strong resistance to high resistance. The simulation results show that the algorithm can correctly partition internal and external faults, and is not affected by fault location, fault type and operation mode of converter station, and has high sensitivity and reliability. Here, a modular multi-level converter (MMC)-high-voltage direct current (HVDC) AC side transmission line protection scheme based on positive sequence fault component current ratio is proposed. The positive sequence fault component current of the MMC-HVDC side is calculated by using the positive sequence fault component current of the power grid side, and the fault is identified through the significant difference between the ratio of the positive sequence fault component current of the MMC-HVDC side and the actual positive sequence fault component current of the MMC-HVDC side in the case of internal and external ***
In this Letter, we propose a method to control large-scale multiagent systems swarming in a ring. Specifically, we use a continuification-based approach that transforms the microscopic, agent-level description of the ...
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In this Letter, we propose a method to control large-scale multiagent systems swarming in a ring. Specifically, we use a continuification-based approach that transforms the microscopic, agent-level description of the system dynamics into a macroscopic, continuum-level representation, which we employ to synthesize a control action towards a desired distribution of the agents. The continuum-level control action is then discretized at the agent-level in order to practically implement it. To confirm the effectiveness and the robustness of the proposed approach, we complement theoretical derivations with numerical simulations.
High-relative-degree control barrier functions play a prominent role in automotive safety and in robotics. In this article, we launch a generalization of this concept for PDE control, treating a specific, physically r...
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High-relative-degree control barrier functions play a prominent role in automotive safety and in robotics. In this article, we launch a generalization of this concept for PDE control, treating a specific, physically relevant model of thermal dynamics where the boundary of the PDE moves due to a liquid-solid phase change-the so-called Stefan model. The familiar quadratic programming (QP) design is employed to ensure safety but with CBFs that are infinite-dimensional (including one control barrier "functional") and with safe sets that are infinite-dimensional as well. Since, in the presence of actuator dynamics, at the boundary of the Stefan system, this system's main CBF is of relative degree 2, an additional CBF is constructed, by backstepping design, which ensures the positivity of all the CBFs without any additional restrictions on the initial conditions. It is shown that the "safety filter" designed in this article guarantees safety in the presence of an arbitrary operator input. This is similar to an automotive system in which a safety feedback law overrides-but only when necessary-the possibly unsafe steering, acceleration, or braking by a vigorous but inexperienced driver. Simulations have been performed for a process in metal additive manufacturing, which show that the operator's heat-and-cool commands to the Stefan model are being obeyed but without the liquid ever freezing.
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