Boundary Control Law Research Articles

Prescribed-time stabilisation for uncertain reaction-diffusion equations with Neumann boundary control

We study the Neumann boundary feedback stabilisation for reaction-diffusion equations with matched boundary uncertainties (including internal uncertainty and external disturbance) within a prescribed time, where the convergence time is independent of initial conditions and can be prescribed a priori. We first estimate the uncertainties within the prescribed time by constructing an estimator and then propose the prescribed-time boundary control law by utilising backstepping transformation with a time-varying kernel. Neumann boundary control is verified to be uniformly bounded.

High-Order Barrier Functions: Robustness, Safety, and Performance-Critical Control

In this article, we propose a notion of high-order (zeroing) barrier functions (HOBFs) that generalizes the concept of zeroing barrier functions and guarantees set forward invariance by checking their higher order derivatives. The proposed formulation guarantees asymptotic stability of the forward invariant set, which is highly favorable for robustness with respect to model perturbations. No forward completeness assumption is needed in our setting in contrast to existing HOBF methods. For the case of controlled dynamical systems, we relax the requirement of uniform relative degree and propose a singularity-free control scheme that yields a locally Lipschitz control signal and guarantees safety. Furthermore, the proposed formulation accounts for “performance-critical” control: it guarantees that a subset of the forward invariant set will admit any existing, bounded control law while still ensuring forward invariance of the set. Finally, a nontrivial case study with rigid-body attitude dynamics and interconnected cell regions as the safe region is investigated.

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.

Observer and boundary output feedback control for coupled ODE-transport PDE

This work focuses on the observer design for a first order ODE-transport PDE coupled at the boundary points. A novel anti-collocated observer and an output feedback boundary control law are designed for an under-actuated coupled system using the backstepping method. The homo-directional hyperbolic type PDE is considered with in-domain coupling between the states. The stabilization of the coupled systems are discussed by Lyapunov theory and linear matrix inequality (LMI) approach is implemented to design the gains. The obtained results show that the observer value coincides with the actual ones and it has been demonstrated through numerical examples. The effectiveness of the output feedback controller is also illustrated.

Asynchronous boundary stabilization for T-S fuzzy Markov jump delay reaction-diffusion neural networks

This paper deals with the exponential boundary stabilization for a class of Markov jump reaction-diffusion neural networks (MJRDNNs) with mixed time-varying delays, which is described by T-S fuzzy model. It is assumed that observed modes in boundary controller are not synchronized with the system modes. Based on a hidden Markov model (HMM), a novel asynchronous boundary control law is developed by using observed modes. Compared with the existing control strategies for distributed parameter systems, the asynchronous boundary control scheme can not only save the cost of the controller installation, but also bring less conservativeness. A delay-dependent sufficient condition to guarantee the exponentially mean square stability is established for T-S fuzzy MJRDNNs with mixed time-varying delays by constructing a Lyapunov functional and utilizing the vector-value Wirtinger-type inequality. Meanwhile, in order to get the designing scheme of the boundary controller, an equivalent LMI-based sufficient criterion is established. In the end, the effectiveness of the proposed results is illustrated by simulation examples.

Results on Boundary Control for Parabolic Systems Using Backstepping Method

This paper focuses on the stabilization problem for the linear parabolic system using the backstepping method. The exponentially stability results for considered parabolic system are derived in two cases with Dirichlet and Neumann local terms. Also, the boundary conditions for the problem is assumed to be mixed or Robin-type boundary conditions. The main aim is to achieve the stability of the considered system using the backstepping method with help of Volterra integral transformation. The explicit solutions of kernel functions in integral transformation is obtained by using Laplace transform and designed a boundary control law to the closed-loop system. Finally, the effectiveness and applicability of the derived results are validated through a single-species pattern generation model.

Boundary stabilization and state estimation of ODE-transport PDE with in-domain coupling

This paper deals with the exponential stabilization of first order ODE-transport PDE coupled at the boundary point. A state feedback boundary control law has been formulated with the help of the backstepping method. The main novelty of this paper is that the stabilization of the coupled system is discussed by Lyapunov theory and the appropriate observer gain is designed by using the linear matrix inequalities (LMIs). An anti-collocated observer design for the corresponding dual system is also presented. The state feedback boundary controller, observer design and the stabilization of the closed-loop system are discussed in detail with illustrative numerical examples.

Simultaneous downstream and upstream output-feedback stabilization of cascaded freeway traffic

In this paper, we develop boundary output feedback control laws for traffic flow on two cascaded freeway segments connected by a junction. The macroscopic traffic dynamics are governed by the Aw–Rascle–Zhang (ARZ) model in which two subsystems of second-order nonlinear partial differential equations (PDEs) describe the evolution of traffic density and velocity on each segment. Due to the change of road access at the junction, different equilibria are considered for the two connected segments. To suppress stop-and-go traffic oscillations on the cascaded roads, we consider a ramp metering that regulates the traffic flow rate entering from the on-ramp to the mainline freeway. Different control designs are proposed such that the output feedback stabilization is realized with either the ramp metering located at the middle junction or the outlet with only boundary measurements of flow rate and velocity. The control objective is to simultaneously stabilize the upstream and downstream traffic to a given spatially-uniform constant steady-state. The distinct actuation locations motivate our design of two different delay-robust full state feedback control laws. The proposed designs are based on the PDE backstepping methodology and guarantee the exponential stability of the under-actuated network of two systems of two hyperbolic PDEs. Collocated boundary observers are proposed to construct output feedback controllers. Numerical simulations are performed to validate the control designs. Stabilization performance of the two output feedback is compared and evaluated with Proportional Integral (PI) boundary feedback controllers. Robustness to delays is also investigated.

Observer-based output feedback control design for a fractional ODE and a fractional PDE cascaded system

This paper considers a new bi-directional cascaded system of a fractional ordinary differential equation (FODE) and a fractional partial differential equation (FPDE) which interacts at an intermediate point. The space-dependent coefficients, interaction between the FODE and FPDE at an intermediate point and the presence of fractional calculus makes the FODE–FPDE cascaded system, representative. In this note, we first apply an invertible integral transformation to convert the system into a FODE–FPDE coupled system, as the target system, which is Mittag–Leffler stable. Using the backstepping method and under some assumptions of the space-dependent coefficients, we work out the kernel functions in the transformation and we design a boundary controller. Also, by the invertibility of the transformation, we show the Mittag–Leffler stability of the closed-loop system via the Lyapunov approach. Second, we propose an observer for which we prove that it can well estimate the original cascaded system. Then, we design an output feedback boundary control law and show that the closed-loop system is Mittag–Leffler stable under the designed output feedback control law. Finally, we present some numerical illustrations to show the correctness of the theoretical results.

Boundary control of a vertical nonlinear flexible manipulator considering disturbance observer and deflection constraint with torque and boundary force feedback signals

In this paper, boundary control (BC) laws are designed to find a BC solution for a single-link nonlinear vertical manipulator to suppress the link’s transverse vibrations and control the rigid body nonlinear large rotating motion. The governing equations of motions and boundary conditions, which all consist of a set of PDEs and ODEs have been derived based on the Hamilton principle. It is desired to regulate large angular orientation, suppress the flexible link’s transverse vibrations and compensate the boundary disturbance simultaneously. The amount of elastic boundary vibration has remained within the constraint range. By considering novel Barrier-Integral Lyapunov functional in order to prevent the amount of boundary deflection from violating the constraints and avoiding any simplifications, in the presence of external boundary disturbance, proper control feedbacks and boundary disturbance observer are adopted in order to reach control objectives and to compensate external boundary disturbance effects simultaneously. By choosing proper boundary feedback, system states are proven to be uniform ultimate bounded and converge exponentially toward a small neighbourhood of zero. In order to illustrate the performance of the proposed control approach, numerical simulation results are provided.

PDE-Based Modeling and Non-collocated Feedback Control of Electrosurgical-Probe/Tissue Interaction.

The first control-oriented model of the interaction of an electrosurgical probe with organic tissue, based on a 1-D PDE with a moving boundary, is introduced. To attain the desired electrosurgically-induced tissue changes using this model, a non-collocated output feedback moving boundary control law is proposed. The latter is realized through a novel non-collocated pointwise temperature-based state observer for the two-phase Stefan problem. Simulation demonstrates that the controller proposed meets the performance objective. The controller implementation is also discussed.

Design bounded robust controller using HJB solution for the nonlinear hybrid dynamical systems

This paper deals with the bounded robust controller design for event wise multiple models of the nonlinear hybrid dynamical systems (NHDS) with matched and unmatched uncertainties. The Hamilton-Jacobi-Bellman (HJB) equation based bounded robust control problem is proposed using a suitable non-quadratic term in the objective function of event wise local linearized model. The families of local bounded control laws are designed, to enforce stability using Lyapunov theorem for the individual events with the knowledge of the maximum bound of uncertainty. Using direct method of Lyapunov stability, the local bounded control law is shown to be optimal with respect to the objective function of the system under uncertainty. The new generalized Lyapunov function is proposed with crisp logic variables to establish the stability around the equilibrium points using bounded robust control law. Simulation and experimental results are presented, which validates the proposed bounded robust control approach.

Adaptive constraint control for flexible manipulator systems modeled by partial differential equations with dead‐zone input

SummaryIn this article, based on partial differential equations (PDEs), the flexible manipulator system with both dead‐zone input and state constraints is studied. The dynamic model of the flexible manipulator system is described by PDEs. The parameters of the dead‐zone input are unknown, and the state constraint problem is also considered. An adaptive approach is proposed to offset the effects caused by dead‐zone input. Thus, to guarantee that all states remain within their respective constraint regions, the boundary control law based on the barrier Lyapunov function is given, and an adaptive controller is designed. According to the Lyapunov analysis method, the control method is given to ensure that all signals of the closed‐loop system are bounded and all states satisfy the constraint conditions. Finally, simulation results show the effectiveness of the proposed control method in this article.

A Direct Bounded Control Method for Transient Stability Assessment

Abstract The paper proposes a direct method for transient stability assessment, which is more efficient than traditional time-domain simulation methods. To achieve this objective, a new bounded control law for the model of a power system is designed. This yields superior closed-loop transient performances when compared to those achievable with traditional automatic voltage regulators and power system stabilizers. The designed control law allows defining an energy-based Lyapunov function which is instrumental in assessing transient stability properties of the post-fault system. A case study on a single machine infinite bus power system model is presented to illustrate the merits of the proposed method.

Adaptive Vibration Control for an Active Mass Damper of a High-Rise Building

As a kind of large flexible structure, high-rise buildings need to consider wind-resistant and anti-seismic problems for the safety of occupants and properties, especially in coastal areas. This article proposes an infinite dimensional model and an adaptive boundary control law for an active mass damper (AMD) on this question. The dynamic model of the high-rise building is a combination of some storeys which have flexible walls and rigid floors under a series of physical conditions. Then the adaptive boundary controller is acted on an AMD which is equipped on the top floor, in order to suppress the vibration of every floor and guarantee the comfort of residents. Moreover, simulations and experiments are carried out on a two-floor flexible building to illustrate the effectiveness of the proposed control strategy.

Output feedback control of two-lane traffic congestion

This paper develops output feedback boundary control to mitigate traffic congestion of an unidirectional two-lane freeway segment. The macroscopic traffic dynamics are described by the Aw–Rascle–Zhang (ARZ) model respectively for both the fast and slow lanes. The traffic density and velocity of each of the two lanes are governed by coupled 2 × 2 nonlinear hyperbolic partial differential equations (PDEs). Lane-changing interactions between the two lanes lead to exchanging source terms between the two pairs second-order PDEs. Therefore, we are dealing with 4 × 4 nonlinear coupled hyperbolic PDEs. Based on driver’s preference for the slow and fast lanes, a reference system of lane-specific uniform steady states in congested traffic is chosen. To stabilize traffic densities and velocities of both lanes to the steady states, two distinct variable speed limits (VSLs) are applied at outlet boundary, controlling the traffic velocity of each lane. Using backstepping transformation, we map the coupled heterodirectional hyperbolic PDE system into a cascade target system, in which the traffic oscillations are damped out through actuation of the velocities at the downstream boundary. Two full-state feedback boundary control laws are developed. We also design a collocated boundary observer for state estimation with sensing of the densities at the outlet. Output feedback boundary controllers are obtained by combining the collocated observer and full-state feedback controllers. The finite time convergence to equilibrium is achieved and a performance improvement in fuel consumption, drivers’ comfort and total travel time is demonstrated with the proposed output feedback controllers for some parameter choice, compared with the open-loop system. The proposed output feedback design is also validated for different congested traffic scenarios and compared with one-lane backstepping and PI control designs.

Boundary Control Design for Traffic With Nonlinear Dynamics

We address a problem of boundary control for a nonlinear scalar conservation law. Namely, this article is devoted to the boundary control of a Lighthill-Whitham-Richards (LWR) partial differential equation (PDE) with triangular flux function evolving along a single road. The target state is a time- and space-dependent trajectory. The boundary control law is constructed using the analytical solution of the Hamilton–Jacobi (H–J) equation, which is an integral form of the LWR PDE. We design a feedback controller and illustrate its performance on a numerical example using the Godunov scheme.

Inverse Optimal Boundary Tracking Control and Observer Design for a One-Link Rotating Flexible Arm

Abstract A new design of an inverse optimal boundary control law is first proposed to force a one-link rotating flexible arm with a tip mass subject to bounded disturbances to track a reference model. The proposed boundary control law ensures global practical exponential stability of the tracking error system and is optimal in the sense that it minimizes a cost functional that appropriately penalizes both the tracking errors and the boundary control without having to solve a Hamilton–Jacobi–Bellman equation. Based on the above boundary control development, an observer-control approach is proposed to design a new inverse optimal observer that globally practically exponentially estimates the one-link rotating flexible arm's states using only measurements at the actuated boundary.

Disturbance observer‐based adaptive boundary iterative learning control for a rigid‐flexible manipulator with input backlash and endpoint constraint

SummaryIn this article, an observer‐based adaptive boundary iterative learning control law is developed for a class of two‐link rigid‐flexible manipulator with input backlash, the unknown external disturbance, and the endpoint constraint. To tackle the backlash nonlinearities and ensure the vibration suppression, the disturbance observers based upon the iterative learning conception are considered in the adaptive boundary control design. A barrier Lyapunov function is incorporated with boundary control law to restrict the endpoint state. Based on the defined barrier composite energy function, the tracking angle error convergence of the rigid part is guaranteed, and the vibrations of the flexible part are suppressed through the rigorous analysis. Finally, a numerical simulation is provided to illustrate the effectiveness of the proposed control.

Stabilization of an ODE-PDE cascaded system by boundary control

Stabilization of an ODE-PDE cascaded system is considered in this paper. The PDE subsystem consists of a system of heat equations, and a control input is set on a boundary. The ODE subsystem is a normal control system cascaded with the PDE subsystem at another boundary, in which its control input is the heat flow rate of the PDE subsystem at this boundary. A boundary control law is constructed by PDE backstepping procedure and lots of complicated mathematical computations. Experiential stability of the closed loop is achieved under this control law.