arXiv subjects

Guchuan Zhu

Publications and source records attributed to Guchuan Zhu.

29 records · Page 2Linked to original sources

arXiv2020

A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances

In this paper, we introduce a weak maximum principle-based approach to input-to-state stability (ISS) analysis for certain nonlinear partial differential equations (PDEs) with boundary disturbances. Based on the weak maximum principle, a classical result on the maximum estimate of solutions to linear parabolic PDEs has been extended, which enables the ISS analysis for certain {}{nonlinear} parabolic PDEs with boundary disturbances. To illustrate the application of this method, we establish ISS estimates for a linear reaction-diffusion PDE and a generalized Ginzburg-Landau equation with {}{mixed} boundary disturbances. Compared to some existing methods, the scheme proposed in this paper involves less intensive computations and can be applied to the ISS analysis for a {wide} class of nonlinear PDEs with boundary disturbances.

math.AP↗

arXiv2019

Aggregate Power Control of Heterogeneous TCL Populations Governed by Fokker-Planck Equations

This paper addresses the modeling and control of heterogeneous populations of thermostatically controlled loads (TCLs) operated by model predictive control (MPC) schemes at level of each TCL. It is shown that the dynamics of such TCLs populations can be described by a pair of Fokker-Planck equations coupled via the actions in the domain. The technique of input-output feedback linearization is used in the design of aggregate power control, which leads to a nonlinear system in closed loop. Well-posedness analysis is carried out to validate the developed control scheme. The closed-loop stability of the system is assessed by a rigorous analysis. A simulation study is contacted, and the obtained results confirm the validity and the effectiveness of the proposed approach.

math.OC↗

arXiv2019

Input-to-State Stability of a Clamped-Free Damped String in the Presence of Distributed and Boundary Disturbances

This note establishes the Exponential Input-to-State Stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades, establishing ISS properties with respect to boundary disturbances remains challenging. One of the well-known methods for well-posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one. However, the resulting distributed disturbance involves time derivatives of the boundary perturbation. Thus, the subsequent ISS estimate depends on its amplitude, and may not be expressed in the strict form of ISS properties. To solve this problem, we show for a clamped-free damped string equation that the projection of the original system trajectories in an adequate Riesz basis can be used to establish the desired EISS property.

math.OC↗

arXiv2019

Boundary Control of a Nonhomogeneous Flexible Wing with Bounded Input Disturbances

This note deals with the boundary control problem of a nonhomogeneous flexible wing evolving under unsteady aerodynamic loads. The wing is actuated at its tip by flaps and is modeled by a distributed parameter system consisting of two coupled partial differential equations. Based on the proposed boundary control law, the well-posedness of the underlying Cauchy problem is first investigated by resorting to the semigroup theory. Then, a Lyapunov-based approach is employed to assess the stability of the closed-loop system in the presence of bounded input disturbances.

math.OC↗

arXiv2018

ISS with Respect to Boundary and In-domain Disturbances for a Coupled Beam-String System

This paper addresses the robust stability of a boundary controlled system coupling two partial differential equations (PDEs), namely beam and string equations, in the presence of boundary and in-domain disturbances under the framework of input-to-state stability (ISS) theory. Well-posedness assessment is first carried out to determine the regularity of the disturbances required for guaranteeing the unique existence of the solution to the considered problem. Then, the method of Lyapunov functionals is applied in stability analysis, which results in the establishment of some ISS properties {}{with respect to} disturbances. As the analysis is based on the \emph{a priori} estimates of the solution to the PDEs, it allows avoiding the invocation of unbounded operators while obtaining the ISS gains in their original expression without involving the derivatives of boundary disturbances.

math.AP↗

arXiv2018

A De Giorgi Iteration-based Approach for the Establishment of ISS Properties for Burgers' Equation with Boundary and In-domain Disturbances

This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De~Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in $L^2$-norm for Burgers' equation have been established using this method. Moreover, as an application of De~Giorgi iteration, ISS in $L^\infty$-norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-$D$ {linear} {unstable reaction-diffusion equation} have also been established. It is the first time that the method of De~Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear {partial differential equations (PDEs)

math.AP↗

arXiv2018

Boundary feedback stabilization of a flexible wing model under unsteady aerodynamic loads

This paper addresses the boundary stabilization of a flexible wing model, both in bending and twisting displacements, under unsteady aerodynamic loads, and in presence of a store. The wing dynamics is captured by a distributed parameter system as a coupled Euler-Bernoulli and Timoshenko beam model. The problem is tackled in the framework of semigroup theory, and a Lyapunov-based stability analysis is carried out to assess that the system energy, as well as the bending and twisting displacements, decay exponentially to zero. The effectiveness of the proposed boundary control scheme is evaluated based on simulations.

math.OC↗

arXiv2017

Input-to-State Stability with Respect to Boundary Disturbances for a Class of Semi-linear Parabolic Equations

This paper studies the input-to-state stability (ISS) properties based on the method of Lyapunov functionals for a class of semi-linear parabolic partial differential equations (PDEs) with respect to boundary disturbances. In order to avoid the appearance of time derivatives of the disturbances in ISS estimates, some technical inequalities are first developed, which allow directly dealing with the boundary conditions and establishing the ISS based on the method of Lyapunov functionals. The well-posedness analysis of the considered problem is carried out and the conditions for ISS are derived. Two examples are used to illustrate the application of the developed result.

math.OC↗

arXiv2017

A De Giorgi Iteration-based Approach for the Establishment of ISS Properties of a Class of Semi-linear Parabolic PDEs with Boundary and In-domain Disturbances

This paper addresses input-to-state stability (ISS) properties with respect to boundary and in-domain disturbances for a class of semi-linear partial differential equations (PDEs) subject to Dirichlet boundary conditions. The developed approach is a combination of the method of De Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS in $L^\infty$-norm for a 1-$D$ transport equation and the ISS in $L^2$-norm for Burgers' equation have been established using this method. As an application of the main result for the 1-D transport equation, a study on ISS properties in $L^\infty$-norm of a 1-D transport equation with anti-stable term under boundary feedback control is carried out. This is the first time that the method of De Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and be applied to a wider class of nonlinear parabolic PDEs.

math.OC↗

arXiv2015

Flatness-based Deformation Control of an Euler-Bernoulli Beam with In-domain Actuation

This paper addresses the problem of deformation control of an Euler-Bernoulli beam with in-domain actuation. The proposed control scheme consists in first relating the system model described by an inhomogeneous partial differential equation to a target system under a standard boundary control form. Then, a combination of closed-loop feedback control and flatness-based motion planning is used for stabilizing the closed-loop system around reference trajectories. The validity of the proposed method is assessed through well-posedness and stability analysis of the considered systems. The performance of the developed control scheme is demonstrated through numerical simulations of a representative micro-beam.

math.OC↗

arXiv2015

In-domain control of a heat equation: an approach combining zero-dynamics inverse and differential flatness

This paper addresses the set-point control problem of a heat equation with in-domain actuation. The proposed scheme is based on the framework of zero dynamics inverse combined with flat system control. Moreover, the set-point control is cast into a motion planing problem of a multiple-input, multiple-out system, which is solved by a Green's function-based reference trajectory decomposition. The validity of the proposed method is assessed through convergence and solvability analysis of the control algorithm. The performance of the developed control scheme and the viability of the proposed approach are confirmed by numerical simulation of a representative system.

math.OC↗