Partial Differential Equation Ordinary Differential Equation Research Articles

SimLivA–Modeling ischemia‐reperfusion injury in the liver: A first step towards a clinical decision support tool

SummaryThe SIMulation supported LIVer Assessment for donor organs (SimLivA) project aims to develop a mathematical model to accurately simulate the influence of mechanical alterations in marginal liver grafts (specifically steatotic ones) and cold ischemia on early ischemia‐reperfusion injury (IRI) during liver transplantation. Our project tackles significant research challenges, including the co‐development of computational methodologies, experimental studies, clinical processes, and technical workflows. We aim to refine a continuum‐biomechanical model for enhanced IRI prediction, collect pivotal experimental and clinical data, and assess the clinical applicability of our model. Our efforts involve augmenting and tailoring a coupled continuum‐biomechanical, multiphase, and multi‐scale partial differential equation‐ordinary differential equation (PDE‐ODE) model of the liver lobule, allowing us to numerically simulate IRI depending on the degree of steatosis and the duration of ischemia. The envisaged model will intertwine the structure, perfusion, and function of the liver, serving as a crucial aid in clinical decision‐making processes. We view this as the initial step towards an in‐silico clinical decision support tool aimed at enhancing the outcomes of liver transplantation. In this paper, we provide an overview of the SimLivA project and our preliminary findings, which include: a cellular model that delineates critical processes in the context of IRI during transplantation; and the integration of this model into a multi‐scale PDE‐ODE model using a homogenized, multi‐scale, multi‐component approach within the Theory of Porous Media (TPM) framework. The model has successfully simulated the interconnected relationship between structure, perfusion, and function—all of which are integral to IRI. Initial results show simulations at the cellular scale that describe critical processes related to IRI during transplantation. After integrating this model into a multiscale PDE‐ODE model, first simulations were performed on the spatial distribution of key functions during warm and cold ischaemia. In addition, we were able to study the effect of tissue perfusion and temperature, two critical parameters in the context of liver transplantation and IRI.

Event-Triggered Underactuated Control for Networked Coupled PDE-ODE Systems With Singular Perturbations

An improved fuzzy event-triggered underactuated control strategy for networked coupled partial differential equation-ordinary differential equation (PDE-ODE) systems subject to singular perturbations is established. First, the networked coupled PDE-ODE systems are combined into an integrated system by representing the ODE and PDE states as a new state variable. Subsequently, a singular perturbed matrix is introduced to describe the multiple time-scale phenomenon of the system. In addition, an improved fuzzy controller is proposed to ensure the stability of the closed-loop coupled PDE-ODE systems based on boundary measurements. The novelties of the designed control strategy pertain to the following aspects: 1) The dynamic event-triggered mechanism is optimized; 2) an underactuated controller is proposed; and 3) the controller is constructed in a unified form. Finally, a hypersonic rocket vehicle's simulation is performed to demonstrate the method's effectiveness.

Dynamic stabilization for a cascaded beam PDE–ODE system with boundary disturbance

We are concerned with the dynamic stabilization for a cascaded Euler–Bernoulli beam (EBB) partial differential equation (PDE)–ordinary differential equation (ODE) system subject to boundary control and matched internal uncertainty and external disturbance. State feedback stabilization of such system without disturbance has been recently discussed by X.H. Wu, H. Feng (Sci China Inf Sci, 2022, 65(5): 159202). An infinite‐dimensional disturbance estimator is constructed in order to estimate the total disturbance. By compensating the total disturbance, we design a state observer to trace the state and then an estimated state and estimated total disturbance‐based output feedback control law. It is proved that the original system is exponentially stable, and other states of the closed‐loop are bounded. Some numerical simulations are presented.

Boundary output feedback stabilization for a cascaded wave PDE-ODE system with velocity recirculation and matched disturbance

In this paper, we are concerned with a boundary stabilization problem of a cascaded wave PDE-ODE (partial differential equation-ordinary differential equation) system with velocity recirculation and matched disturbance. First, an auxiliary PDE system is proposed to estimate the boundary disturbance by active disturbance rejection control (ADRC) method. Next with the estimation of the disturbance in hand, a state observer is constructed to recover the state of the PDE-ODE system. Then we design an output feedback controller under which the solution of the resulting closed-loop system is bounded and the state of the cascaded wave PDE-ODE system decays exponentially. Finally, some simulation examples are given to illustrate the effectiveness of the designed controller.

Stability analysis of reaction–diffusion PDEs coupled at the boundaries with an ODE

This paper addresses the derivation of generic and tractable sufficient conditions ensuring the stability of a coupled system composed of a reaction–diffusion partial differential equation (PDE) and a finite-dimensional linear time invariant ordinary differential equation (ODE). The coupling of the PDE with the ODE is located either at the boundaries or in the domain of the reaction–diffusion equation and takes the form of the input and output of the ODE. We investigate boundary Dirichlet/Neumann/Robin couplings, as well as in-domain Dirichlet/Neumann couplings. The adopted approach relies on the spectral reduction of the problem by projecting the trajectory of the PDE into a Hilbert basis composed of the eigenvectors of the underlying Sturm–Liouville operator and yields a set of sufficient stability conditions taking the form of LMIs. We propose numerical examples, consisting of an unstable reaction–diffusion equation and an unstable ODE, such that the application of the derived stability conditions ensure the stability of the resulting coupled PDE–ODE system.

Adaptive Neural Network Control of an Uncertain Nuclear Refueling Machine With Input Backlash and Asymmetric Output Constraint

This paper studies the control problem of an uncertain nuclear refueling machine (RM) system in the presence of uncertainty, external disturbance, input backlash and asymmetric output constraint. Taking into account the effects of external disturbance and input backlash, the RM system with fuel rod is modeled as a copling partial differential equation-ordinary differential equation (PDE-ODE). Using the backstepping method, an adaptive neural network control scheme is designed to drive the RM and bridge to the desired positions and simultaneously reduce the vibration of fuel rod. A novel asymmetric tangent-type barrier Lyapunov function (Tan-BLF) is constructed to restrict the position tracking error into the given range. The formulation of backlash nonlinearity is transformed into the expected input and nonlinear error. Then the combination of input backlash error and boundary disturbance is defined as a disturbance-like item. Applying the robust control strategy and adaptive technique, two auxiliary input signals are proposed to offset the impact of the disturbance-like item. The adaptive neural network (NN) is employed to compensate for the system uncertainty. The practical stability of the RM system with the proposed control is demonstrated via the theoretical Lyapunov analysis. Simulation verifies the performance of the designed control scheme.

Phase-field simulation of crack propagation in quasi-brittle materials: COMSOL implementation and parameter sensitivity analysis

Recently, the use of the phase-field method (PFM) to simulate the fracture process of brittle materials has attracted increasing attention. The PFM describes the fracture process through a series of differential equations, thus avoiding tedious crack surface tracking and offering advantages in simulating crack initiation, propagation, and bifurcation. The essence of the PFM is a multifield coupling problem, so it is supposed that the COMSOL Multiphysics commercial finite element software, which is particularly suitable for solving multifield coupling problems, should be more efficient and simpler to implement for the PFM. In this paper, a crack propagation model for quasi-brittle materials based on PFM is implemented in COMSOL Multiphysics by means of the solid mechanics module and secondary development interfaces of the partial differential equation (PDE), domain ordinary differential equation (ODE) and differential algebraic equation (DAE). Combined with the collected tensile and shear numerical simulation data, validation studies are carried out both qualitatively and quantitatively. In addition, considering that the PFM involves many parameters would create a significant amount of work for model calibration.Therefore, multifactor sensitivity analysis based on the orthogonal test method is used to identify the parameter' sensitivity. The results show that the use of the solid mechanics module and interfaces of the PDE and domain ODE and DAE are effective for phase-field modelling, and the proposed method could reasonably characterize the whole fracture process of quasi-brittle materials. The sensitivity analysis results revealed that Young's modulus (E) and critical energy release rate (G c) are the main factors affecting the output results of the model.

Regulation‐triggered adaptive control of a hyperbolic PDE‐ODE model with boundary interconnections

SummaryWe present a certainty equivalence‐based adaptive boundary control scheme with a regulation‐triggered batch least‐squares identifier, for a heterodirectional transport partial differential equation‐ordinary differential equation (PDE‐ODE) system where the transport speeds of both transport PDEs are unknown. We use a nominal controller which is fed piecewise‐constant parameter estimates from an event‐triggered parameter update law that applies a least‐squares estimator to data “batches” collected over time intervals between the triggers. A parameter update is triggered by an observed growth in the norm of the PDE state. The proposed triggering‐based adaptive control guarantees: (1) the absence of a Zeno phenomenon; (2) parameter estimates are convergent to the true values in finite time (from most initial conditions); (3) exponential regulation of the plant states to zero. The effectiveness of the proposed design is verified by a numerical example.

Modeling and Control of a Quadrotor UAV Equipped With a Flexible Arm in Vertical Plane

In the field of unmanned aerial vehicles (UAVs), aerial manipulations are receiving considerable attention because of their potential application to tasks such as pick and place, detection, and inspection. However, short flight endurance times and concerns about the safety to surroundings during interacting heavily limit the expansion of aerial manipulations in real implementations. To overcome these challenges, this paper focuses on a system in which a quadrotor UAV is equipped with a lightweight and flexible arm. Based on the infinite-dimensional dynamics, the mathematic model of system is described by a hybrid partial differential equation-ordinary differential equation (PDE-ODE). An easily implementable controller is derived from a Lyapunov functional construction related to the energy of the system. The proposed controller ensures global Lyapunov stability for nonlinear system and local asymptotic stability for the linearized system. Further, it is shown that the proposed controller realizes stable motion of the aerial manipulator as well as vibration control of the flexible arm. Finally, numerical simulations are conducted to investigate the validity of the proposed controller.

Grasp and Orientation Control of an Object by Two Euler–Bernoulli Arms With Rolling Constraints

Abstract This paper focuses on grasping and manipulation of an object by two one-link flexible arms. By taking rolling constraints between the arm tip and the grasped object, the arms have the potential to grasp and manipulate an object at the same time. To realize grasping and manipulation by two flexible arms, a boundary controller is derived from a Lyapunov functional related to the total energy of a dynamic model described by a hybrid partial differential equation-ordinary differential equation (PDE-ODE) model. The derived controller consists of the bending moment at the root of the arm, the rotational angle, and the angular velocity of the motor. In particular, the controller does not need the feedback of the information of the grasped object, and thus, it is easy to implement the controller. Further, it is shown that the derived controller realizes stable grasping and orientation control of the object as well as vibration control of the arms. Finally, experiments and numerical simulations are conducted to investigate the validity of the derived boundary controller.

Exponential stability of dual flexible arms for grasping and orientation control

The stability of two one-link flexible arms for grasping and orientation control of an object is studied. Flexible arms are modelled by Euler–Bernoulli beam model and the overall system is represented by a hybrid partial differential equation (PDE)–ordinary differential equation (ODE) model. The authors' primary concern is the stability analysis of this hybrid PDE–ODE model. In particular, they use the frequency domain method and prove the exponential stability of this system under their previously proposed boundary controller. In addition, they discuss the robustness of the closed-loop system with respect to the several disturbances including disturbances distributed over the arms and disturbances acting at boundaries. Finally, simulation results are presented to investigate the theoretical results.

Bounded bilinear control of coupled first-order hyperbolic PDE and infinite dimensional ODE in the framework of PDEs with memory

In this work, we consider the problem of bounded bilinear tracking control of a system of coupled first-order hyperbolic partial differential equation (PDE) with an infinite dimensional ordinary differential equation (ODE). This coupled PDE-infinite ODE system can be viewed as a degenerate system of two coupled first-order hyperbolic PDEs, the velocity of the ODE part vanishing. First, we convert this PDE-infinite ODE system into a first-order hyperbolic PDE with memory and investigate the bounded bilinear control problem in this framework. We consider as manipulated variable the constrained wave propagation velocity, which makes the control problem bounded and bilinear, and we take the measurements at the boundaries. To account for the actuator's constraints, we develop conditions under which the bounded control law ensures stability and tracking performances. This leads to a specification of the state-space region that enforces the desired system's closed-loop behaviour. To overcome the lack of full-state measurements, we design an observer-based bounded output-feedback control law which guarantees the reference tracking and uniform asymptotic stability of the system in closed-loop. A strong motivation of our work is the control problem of the solar collector parabolic trough where the manipulated control variable (the pump volumetric flow rate) is bilinear with respect to the PDE-infinite ODE model, and the measurements are taken at the boundary (tube's outlet). Simulation results illustrate the efficiency of the proposed control strategy.

Input-to-state stability of coupled hyperbolic PDE-ODE systems via boundary feedback control

We herein investigate the boundary input-to-statestability (ISS) of a class of coupled hyperbolic partial differential equation-ordinary differential equation (PDE-ODE) systemswith respect to the presence of uncertainties and externaldisturbances. The boundary feedback control of the proportional typeacts on the ODE part and indirectly affects the hyperbolic PDEdynamics via the boundary input. Using the strict Lyapunovfunction, some sufficient conditions in terms of matrix inequalitiesare obtained for the boundary ISS of the closed-loop hyperbolicPDE-ODE systems. The feedback control laws are designed by combiningthe line search algorithm and polytopic embedding techniques.The effectiveness of the designed boundary control is assessed byapplying it to the system of interconnected continuous stirred tankreactor and a plug flow reactor through a numericalsimulation.

Compensation of Transport Actuator Dynamics With Input-Dependent Moving Controlled Boundary

We introduce and solve the stabilization problem of a transport partial differential equation (PDE)/nonlinear ordinary differential equation (ODE) cascade, in which the PDE state evolves on a domain whose length depends on the boundary values of the PDE state itself. In particular, we develop a predictor-feedback control design, which compensates such transport PDE dynamics. We prove local asymptotic stability of the closed-loop system in the $C^1$ norm of the PDE state employing a Lyapunov-like argument and introducing a backstepping transformation. We also highlight the relation of the PDE–ODE cascade to a nonlinear system with input delay that depends on past input values and present the predictor-feedback control design for this representation as well.

Axial Vibration Suppression in a Partial Differential Equation Model of Ascending Mining Cable Elevator

Lifting up a cage with miners via a mining cable causes axial vibrations of the cable. These vibration dynamics can be described by a coupled wave partial differential equation-ordinary differential equation (PDE-ODE) system with a Neumann interconnection on a time-varying spatial domain. Such a system is actuated not at the moving cage boundary, but at a separate fixed boundary where a hydraulic actuator acts on a floating sheave. In this paper, an observer-based output-feedback control law for the suppression of the axial vibration in the varying-length mining cable is designed by the backstepping method. The control law is obtained through the estimated distributed vibration displacements constructed via available boundary measurements. The exponential stability of the closed-loop system with the output-feedback control law is shown by Lyapunov analysis. The performance of the proposed controller is investigated via numerical simulation, which illustrates the effective vibration suppression with the fast convergence of the observer error.

Stability analysis of coupled torsional vibration and pressure in oilwell drillstring system

ABSTRACTTo address security issues in oilwell drillstring system, the drilling operation handling which is in generally not autonomous but ensured by an operator may be drill bit destructive or fatal for the machine. To control of stick-slip phenomenon, the drillstring control at the right speed taking only the drillstring vibration is not sufficient as the mud dynamics and the pressure change around the drill pipes cannot be neglected. A coupled torsional vibration and pressure model is presented, and the well-posedness problem is addressed. As a Partial Differential Equation-Ordinary Differential Equation (PDE-ODE) coupled system, and in order to maintain a non destructive downhole pressure, we investigate the control stability with and without the damping term in the wave PDE. In terms of, the torsional variable, the downhole pressure, and the annulus pressure, the coupled system equilibrium is shown to be exponentially stable.

Adaptive output-feedback stabilisation for hybrid PDE–ODE systems with uncertain input disturbances

ABSTRACTThis paper is concerned with the adaptive output-feedback stabilisation for a class of hybrid partial differential equation (PDE)–ordinary differential equation (ODE) systems with uncertain input disturbances. Remarkably, in this paper, only two boundary measurements of the considered system are available for feedback. Moreover, the unknown parameters involved in the system are allowed to belong to an unknown interval. These two aspects make the considered system essentially different from those in the closely related literature. Inspired by the existing results, an observer is first introduced to estimate the unmeasured states of the original system. Then, by adaptive technique and backstepping method, an adaptive output-feedback controller is successfully constructed, which guarantees that the entire closed-loop system is asymptotically stable, and moreover, the parameter estimates converge to their own real values ultimately. Besides, by the semigroup approach, the well-posedness of the entire closed-loop system is achieved. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.

Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems

This paper addresses the problem of feedback control design for a class of linear cascaded ordinary differential equation (ODE)–partial differential equation (PDE) systems via a boundary interconnection, where the ODE system is linear time-invariant and the PDE system is described by an Euler–Bernoulli beam (EBB) equation with variable coefficients. The objective of this paper is to design a static output feedback (SOF) controller via EBB boundary and ODE measurements such that the resulting closed-loop cascaded system is exponentially stable. The Lyapunov’s direct method is employed to derive the stabilization condition for the cascaded ODE–beam system, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, in order to compute the gain matrices of SOF controllers, a two-step procedure is presented to solve the BMI feasibility problem via the existing linear matrix inequality (LMI) optimization techniques. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed design method.