Boundary Control Problem Research Articles

Time-Varying Trajectory Tracking Boundary Control of a Flexible Rotation Beam Based on Servomechanism

In this article, we consider the time-varying trajectory tracking boundary control problem for the flexible rotation beam subject to distributed disturbance. A partial differential equation model for the flexible rotation beam system with the coupling of angle and vibration displacement is established by Hamilton principle. By introducing a servo system generated by the reference signal, a boundary controller is proposed to track the desired time-varying trajectory, which is different from the fixed angular position in most existing work. A Lyapunov-based stability analysis is presented to demonstrate that the tracking error system between the servo system and the original system is uniformly ultimately bounded. Based on the infinite dimensional operator theory, it is proved that the closed-loop system is well posed and all signals remain bounded. Numerical simulations verify the correctness of the theory and the effectiveness of the control strategy. And the experiments based on the Quanser platform demonstrate the same result as well.

Output‐feedback boundary adaptive fault‐tolerant control for scalar hyperbolic partial differential equation systems with actuator faults

SummaryThis article studies the output‐feedback adaptive fault‐tolerant boundary control problem for scalar hyperbolic partial differential equation systems with actuator faults. All the coefficients of the controlled plant are unknown, and two types of actuator faults, that is, multiplicative faults and additive faults, are considered simultaneously. For the state estimation problem, two filters are constructed, based on which an observer is obtained to estimate the system state. A parametric model is established for actuator faults, based on which the parameter updating laws of gradient type are then developed to identify actuator faults and to estimate the unknown system coefficients. With the observer and the parameter updating laws, an output‐feedback adaptive fault‐tolerant boundary control law is developed via infinite‐dimensional backstepping method. The boundness of all the signals involved in the control design is guaranteed and the convergence of system states is also confirmed. Finally, the simulation results are given to testify the effectiveness of the proposed control scheme.

Abstract representation of the SMGTJ equation under rough boundary controls: Optimal interior regularity

We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L‐T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.

Estimation Based Adaptive Constraint Control for a Class of Coupled String Systems

Dear editor, In this letter, the boundary tension constraint-based adaptive boundary control problem is investigated for a class of flexible string systems with unknown boundary disturbances and unknown parameters, with the purpose of suppressing transverse and longitudinal vibrations. Firstly, on the basis of the Hamiltonian principle, two partial differential equations for a transverse-longitudinal coupled string with spatiotemporally varying tension are derived. Secondly, the transverse and longitudinal adaptive boundary controllers are designed by constructing Lyapunov functions. A barrier Lyapunov function is adopted to confine the boundary tension within the constraint and disturbance observers are designed to estimate the unknown boundary disturbances. Finally, stability analysis of the closed-loop system is performed. The simulation results further illustrate the effectiveness of the proposed method.

Finite‐time parameter estimation based boundary control for a class of 2×2 hyperbolic partial differential equation systems with uncertain transport speeds

AbstractThis article investigates the boundary control problem for a class of hyperbolic partial differential equation systems with uncertain transport speeds, in which both state feedback and output feedback cases are considered. The least‐square method is utilized to achieve the finite‐time parameter estimation based on the parametric models. During parameter estimation, proper actuation signals are designed via contradiction method to avoid the singular problems. With the estimated parameters, control laws are designed by the backstepping method for both state‐feedback case and output‐feedback case. Finally, simulation studies are provided to illustrate the efficiency of our results.

Robust non-fragile boundary control for non-linear parabolic PDE systems with semi-Markov switching and input quantization

In this paper, the robust boundary control problem for non-linear PDE systems is discussed. Specifically, the PDE under investigation is of parabolic type with semi-Markov jumping signals subject to non-linearities and parameter uncertainties. The main goal of this paper is to devise a non-fragile boundary control law which assures the robust stabilization of the addressed system in spite of gain fluctuations and quantization in its design. In particular, to reduce the burden in the communication medium, the input signals can be quantized before transmitted to the actuator by using logarithmic quantizers. Moreover, by utilizing the Lyapunov stability theory and vector-valued Wirtinger’s inequality, a set of sufficient conditions for robust stabilization of the underlying system is established in the form of linear matrix inequalities (LMIs). Based on the obtained conditions, the existence of the desired controller is confirmed and an explicit form of the controller gains are developed. Eventually, the efficacious of the extracted theoretical results are manifested by virtue of two simulation examples.

Predictor-feedback stabilization of two-input nonlinear systems with distinct and state-dependent input delays

This paper deals with the predictor-feedback stabilization problem of two-input nonlinear systems with distinct and state-dependent input delays. Since each individual channel may induce a different delay, a sequential approach is employed for the construction of two predictors such that the corresponding input delays can be compensated. Concretely, the predictor over the prediction horizon which corresponds to the smaller input delay is first constructed, and the predictor over the remaining time horizon that corresponds to the larger input delay is in turn constructed, by utilizing the first predictor signal as a basis. The feedback controllers are given based on the predictors. Due to the inherent constraint on the rates of the state-dependent input delays, regional stability results are achieved. Specifically, if the system is forward-complete, for the closed-loop nonlinear system, an estimation of the region of attraction is provided, and for the linear system, the exponential stability is guaranteed. The methodology is applied to the bilateral boundary control problem of hyperbolic partial differential equations (PDEs) with a moving boundary, for illustration of the effectiveness of the methodology.

Distributed Adaptive Mittag–Leffler Formation Control for Second-Order Fractional Multi-Agent Systems via Event-Triggered Control Strategy

This brief investigates the Mittag–Leffler formation bounded control problem for second-order fractional multi-agent systems (FMASs), where the dynamical nodes of followers are modeled to satisfy quadratic (QUAD) condition. Firstly, under the undirected communication topology, for the considered second-order nonlinear FMASs, a distributed event-triggered control scheme (ETCS) is designed to realize the global Mittag–Leffler bounded formation control goal. Secondly, by introducing adaptive weights into triggering condition and control protocol, an adaptive event-triggered formation protocol is presented to achieve the global Mittag–Leffler bounded formation. Thirdly, a five-step algorithm is provided to describe protocol execution steps. Finally, two simulation examples are given to verify the effectiveness of the proposed schemes.

Reinforcement Learning Versus PDE Backstepping and PI Control for Congested Freeway Traffic

We develop reinforcement learning (RL) boundary controllers to mitigate stop-and-go traffic congestion on a freeway segment. The traffic dynamics of the freeway segment are governed by a macroscopic Aw-Rascle-Zhang (ARZ) model, consisting of 2 x 2 quasi-linear partial differential equations (PDEs) for traffic density and velocity. The boundary stabilization of the linearized ARZ PDE model has been solved by PDE backstepping, guaranteeing spatial L² norm regulation of the traffic state to uniform density and velocity and ensuring that traffic oscillations are suppressed. Collocated proportional (P) and proportional-integral (PI) controllers also provide stability guarantees for allowable control gains and are always applicable as model-free control options through gain tuning by trial and error, or by model-free optimization. Although these approaches are mathematically elegant, the stabilization result only holds locally and is usually affected by the change of model parameters. Therefore, we reformulate the PDE boundary control problem as an RL problem that pursues stabilization without knowing the system dynamics, simply by observing the state values. The proximal policy optimization (PPO), a neural network-based policy gradient algorithm, is employed to obtain RL controllers by interacting with a numerical simulator of the ARZ PDE. Being stabilization-inspired, the RL state-feedback boundary controllers are compared and evaluated against the rigorously stabilizing controllers in two cases: 1) in a system with perfect knowledge of the traffic flow dynamics and then 2) in one with only partial knowledge. We obtain RL controllers that nearly recover the performance of the backstepping, P, and PI controllers with perfect knowledge and outperform them in some cases with partial knowledge. It must be noted, however, that the RL controllers are obtained by conducting about one thousand episodes of iterative training on a simulation model. This training cannot be performed in a collision-free fashion in real traffic, nor convergence guaranteed when training. Thus, we demonstrate that the RL approach has learning (i.e., adaptation) potential for traffic PDE systems under uncertain and changing conditions, but RL is neither simple nor a fully safe substitute for model-based control in real traffic systems.

Boundary control problem for a hyperbolic equation loaded along one of its characteristics

This paper investigates the unique solvability of the boundary control problem for a one-dimensional wave equation loaded along one of its characteristic curves in terms of a regular solution. The solution method is based on an analogue of the d’Alembert formula constructed for this equation. We point out that the domain of definition for the solution of DE, when the initial and final Cauchy data given on intervals of the same length is a square. The side of the squire is equal to the interval length. The boundary controls are established by the components of an analogue of the d’Alembert formula, which, in turn, are uniquely established by the initial and final Cauchy data. It should be noted that the normalized distribution and centering are employed in the final formulas of sought boundary controls, which is not typical for initial and boundary value problems initiated by equations of hyperbolic type.

Optimal boundary control problems for a hierarchical age-structured two-species model

This paper discusses the optimal control problems for a nonlinear age-structured two-species model, where elder individuals are more competitive than younger ones, and each species is described by a nonlinear integropartial differential equation with a global feedback boundary condition. First, we establish the existence of a unique nonnegative bounded solution by means of frozen coefficients and the fixed-point theorem. More importantly, we discuss the least deviation-cost problem and the most benefit-cost problem. For the least deviation-cost problem, the existence of an optimal strategy is established by means of Ekeland’s variational principle, and the minimum principle is obtained via an adjoint system. Meanwhile, the corresponding results for the most benefit-cost problem are given. In addition, some numerical experiment results are presented to examine the effects of parameters on the optimal policies and indexes.

Boundary control of a coupled Burgers' PDE‐ODE system

AbstractThis article presents a result of stabilization of a coupled partial differential equation (PDE) and ordinary differential equation (ODE) system through boundary control. The PDE is the Burgers' equation, which is a widely considered nonlinear PDE, partially due to its low order and partially due to its structure analogous to the Navier–Stokes equation, which describes fluid dynamics. The controller we employ for stabilizing this system was first developed from the boundary control problem of the corresponding linearized system, based on an infinite‐dimensional backstepping transformation. The stabilization result is achieved using only one boundary measurement and one boundary control. Numerical simulations show the boundary control law can be used to stabilize the system.

Residual-based a posteriori error estimates for the [formula omitted] version of the finite element discretization of the elliptic Robin boundary control problem

Optimal control problems governed by partial differential equations have become a very active and successful research area. So, in this paper, we analyzed a priori and a posteriori error estimates for the hp finite element discretization of elliptic Robin boundary control problems. With the discrete and continuous optimality conditions of the problem, we constructed the error estimators. Based on the residual of the model equations for the coupled state and control approximations, the upper error bound is proved using Scott–Zhang-type quasi interpolation estimates. In order to provide the optimality, lower error bound is shown using some polynomial inverse estimates in weighted Sobolev spaces. Such estimators can be used to construct reliable adaptive methods for optimal control problems.

Boundary control of stochastic Korteweg-de Vries-Burgers equations

The boundary control problem is considered for stochastic Korteweg-de Vries-Burgers equations. First, a boundary controller is proposed, and a criterion is obtained for mean square exponential stability by using the Lyapunov functional method and inequality techniques. Then, when there exist uncertainties in the system parameters, the robust mean square exponential stability is considered, and a sufficient criterion is obtained. Furthermore, if there are also additive noises in the considered system, the H-infinity performance is investigated and a sufficient condition is obtained to ensure the mean square H-infinity performance. Numerical examples illustrate the validity of the theoretical results.

Control of String Vibrations by Displacement of One End with the Other End Fixed, Given the Deflection Form at an Intermediate Moment of Time

We consider a boundary control problem for the equation of string vibration with given initial and final conditions, given the deflection form at an intermediate moment of time. The control is carried out by displacement of one end with the other end fixed. The problem is reduced to the problem of a distributed action control with zero boundary conditions. We propose a constructive approach to constructing a boundary control action by the separation of variables and methods of the theory of control of finite-dimensional systems. The approach is applied to given functions. A computational experiment was carried out with the construction of the corresponding graphs and their comparative analysis. They confirm the obtained results.

Event-triggered boundary stabilization for coupled semilinear reaction–diffusion systems with spatially varying coefficients

This paper aims to address the event-triggered Robin boundary control problem for exponential stabilization of the coupled semilinear reaction–diffusion systems with spatially varying coefficients. The main used method is the backstepping, which allows us to explicitly give the boundary control formulae. More precisely, we first explore the existence and uniqueness of classical solutions for the considered problem. After this, we propose an event-triggered boundary feedback control law to exponentially stabilize the system under consideration with the Zeno phenomenon being excluded. A numerical result is finally included to illustrate the efficiency of our designed controller.

Crank–Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problems

In this paper, we adopt the optimize‐then‐discretize approach to solve parabolic optimal Dirichlet boundary control problems. First, we derive the first‐order necessary optimality system, which includes the state, co‐state equations, and the optimality condition. Then, we propose Crank–Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second‐order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible, and second‐order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present several numerical examples to validate our theoretical analysis.

Local a posteriori error estimates for boundary control problems governed by nonlinear parabolic equations

We derive local a posteriori error estimates of finite element approximation to nonlinear parabolic boundary control problems in a bounded convex domain with Lipschitz boundary. We apply piecewise linear and continuous finite elements to approximate the state and co-state variables, and the control variable is approximated by using the piecewise constant functions. The discrete-in-time scheme is based on the backward Euler method. We derive a reliable type local a posteriori error estimates for Neumann boundary control problems in the L2([0,T];L2(∂Ω))-norm. The main feature of our estimators is that they are of local character in the sense that the leading terms of the estimators depend on the small neighbourhood of the boundary. These new local a posteriori error bounds can be used to study the behaviour of the state and co-state variables near the boundary. The derived error indicators will provide the necessary feedback for the adaptive mesh refinements in the finite element method. We report numerical tests to illustrate the effectiveness of our error indicators.

Asymptotic Expansion for the Solution of an Optimal Boundary Control Problem in a Doubly Connected Domain with Different Control Intensity on Boundary Segments

Asymptotic Expansion for the Solution of an Optimal Boundary Control Problem in a Doubly Connected Domain with Different Control Intensity on Boundary Segments

Analysis and Approximations of Dirichlet Boundary Control of Stokes Flows in the Energy Space

We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the $L^2$ norm and an energy space seminorm. We prove well-posedness and regularity results for both problems, develop finite element discretizations for both problems, and prove finite element error estimates for the latter problem. The motivation to study the energy space problem follows from our analysis: we prove that the choice of the control space ${\bm L}^2(\Gamma)$ can lead to an optimal control with discontinuities at the corners, even when the domain is convex. We observe this phenomenon in numerical experiments. This behavior does not occur in Dirichlet boundary control problems for the Poisson equation on convex polygonal domains, and may not be desirable in real applications. For the energy space problem, we derive the first order optimality conditions, and show that the solution of the control problem is more regular than the solution of the problem with the ${\bm L}^2(\Gamma)$ regularization. We also prove a priori error estimates for the control in the energy norm, and present several numerical experiments for both control problems on convex and nonconvex domains.