Bounded State Feedback Research Articles

Sliding mode control to stabilization of coupled time fractional parabolic PDEs subject to disturbances

AbstractIn this article, the stabilization of coupled time fractional parabolic partial differential equations subject to external disturbances is investigated. By using sliding mode control method and backstepping approach, a boundary state feedback controller is designed to reject the matched disturbance and achieve the Mittag‐Leffler input‐to‐state stability of closed‐loop system. The existence of the generalized solution to the closed‐loop system is proven by Galerkin approximation scheme. Simulations are presented to illustrate the validity of our theoretical results.

Iterative learning control of an Euler-Bernoulli beam with time-varying boundary disturbance

This paper investigates the stabilization of Euler-Bernoulli beam systems modeled by a partial differential equation (PDE) with unknown time-varying disturbance. An iterative learning controller is designed using only boundary state feedback to realize the vibration control subject to unknown boundary disturbance. The well-posedness for the closed-loop system is given by the operator semigroup theory. Furthermore, the exponentially stable for the closed-loop system is proved by the Lyapunov method. The comparisons with existing results are made to demonstrate the effectiveness and advantages of the proposed boundary iterative learning control method.

State feedback controller design of fractional ordinary differential equations coupled with a fractional reaction–advection–diffusion equation with delay

The boundary control problem of fractional ordinary differential equations coupled with a time fractional reaction–advection–diffusion equation with delay is studied in this paper. To ensure the asymptotic stability of the system we studied, a state feedback boundary controller is proposed. By backstepping method, we transform the fractional coupled system into a chosen target system under a controller. Furthermore, we obtain the existence and uniqueness of the state solution of the considered system. A Lyapunov functional is constructed to show the asymptotic stability of the fractional coupled system by the special fractional Halanay inequality. The asymptotic stability criterion of the fractional coupled system is described by Linear Matrix Inequalities (LIMs). Which can be easily solved and verified. Finally, the applicability of our theoretical results is showed by a numerical simulation.

Local exponential stabilization of a coupled ODE-Fisher’s PDE system

In this paper, we investigate the problem of exponential stabilization of a coupled control system consisting of a linear ordinary differential equation (ODE) and a nonlinear partial differential equation (PDE). The nonlinear PDE is the one-dimensional Fisher’s equation defined on a bounded domain. The control input for the whole system acts at the right boundary of the PDE domain, whereas the left boundary injects a Dirichlet term in the ODE subsystem and the coupling is given as a Robin condition at the left boundary. Based on the infinite dimensional backstepping method for the linear part, a state feedback boundary controller is explicitly derived. Under this controller, we show that the considered coupled ODE-Fisher’s PDE system is globally well-posed in the Hilbert space Rn×L2(0,1). Moreover, using a well-adapted strict Lyapunov functional, we establish its local exponential stabilization around its zero equilibrium in the topology of Rn×L2(0,1)-norm. A simulation result that reflects the effectiveness of the proposed approach is given.

Active disturbance rejection control to stabilization of coupled delayed time fractional-order reaction–advection–diffusion systems with boundary disturbances and spatially varying coefficients

In this paper we consider the boundary asymptotic stabilization of a coupled time fractional-order reaction–advection–diffusion (FRAD) system with boundary input disturbances, spatially varying coefficients and time varying delays. The complex system composition makes it difficult to analyze system property directly. For this issue, we here address it by introducing two transformations to convert the original system into the one where theories of fractional evolution equations and operator semigroup are easily applicable. The active disturbance rejection control (ADRC) and backstepping control are also utilized in investigation. Using ADRC, the disturbance is first estimated by constructing two auxiliary systems and then canceled through an approximation in the feedback-loop. For such auxiliary systems, one is used to take disturbances from the original system and to put them into a stable system. Another is used to estimate disturbances. In the second part, we use backstepping to develop a boundary state feedback control law with disturbance approximation to ’eliminate’ the disturbances and to achieve the closed-loop stability. The well-posedness and disturbance estimation are established by theories of fractional evolution equations. With fractional Halanay’s inequality, sufficient conditions are obtained to make the controlled system asymptotically stable and the auxiliary system bounded. Fractional simulation scheme is constructed to test the proposed synthesis generated by ADRC when the explicit solution of kernel equations does not exist.

Reduced-Order Min-Max Guaranteed Cost Control for Finite-Time Boundedness of Uncertain Linear Systems Using Output Feedback

This letter considers a problem of designing reduced-order dynamic output feedback controllers for finite-time boundedness of uncertain linear time-varying systems subjected to exogenous disturbances. Differential linear matrix inequality conditions are derived for finite-time boundedness with output feedback, such that the uncertain system simultaneously satisfies finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathscr {L}_{2}$ </tex-math></inline-formula> gain in the presence of worst-case exogenous disturbances. The designed min-max controller minimizes an upper bound on the considered performance index, providing a guaranteed cost for disturbance attenuation performance. Finite-time bounded state feedback controller design is shown to be a special case of the proposed method. The performance efficacy and applicability of the controller designed using the proposed iterative algorithm are compared with some of the recent methods from the literature.

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 Controller Design for a Class of Horizontal Belt Transmission System with Boundary Vibration Constraint

In this paper, the problem of transverse vibration suppression of a belt system moving in the horizontal direction is investigated. This system is characterized by the boundary vibration constraint and is affected by external disturbances. For it, we introduced a logarithmic function in the candidate term of the Lyapunov function and used a symbolic function in the controller to compensate for the effects of boundary vibration constraints and boundary disturbances, respectively. In order to better achieve the control objective, we designed a boundary control scheme. The state feedback boundary controller was designed using the boundary signals of the system when they can be available directly. Considering the presence of noise in the practical system, some system signals cannot be measured accurately. Therefore, a high-gain observer was introduced to estimate these signals, and an output feedback boundary controller was designed. Finally, the simulation example showed that both controllers guarantee effective suppression of the transverse vibration of the system without violating the boundary vibration constraints.

Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation

&lt;p style='text-indent:20px;'&gt;In this paper, we solve the problem of rapid exponential stabilization for coupled nonlinear ordinary differential equation (ODE) and &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 1-d $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; unstable linear heat diffusion. The control acts at a boundary of the heat domain and the heat equation enters in the ODE by Dirichlet connection. We show that the infinite dimensional backstepping transformation introduced recently for stabilization of coupled linear ODE-PDE can deal with a nonlinear ODE and obtain a global stabilization result. Our result is innovative and no similar result can be found in the literature as it combines the three following factors, i) nonlinear term in the ODE subsystem, ii) unstable PDE subsystem and iii) mixed boundary condition. Not only this, the techniques used in this work can provide answers to fundamental questions, such as the stabilization of coupled systems where both subsystems may contain nonlinear terms.&lt;/p&gt;

Distributed Damping Assignment for a Wave Equation in the Port-Hamiltonian Framework

In this paper, we propose a full state-feedback boundary control strategy for a one-dimensional wave-like equation with spatially varying parameters and indefinite damping coefficient. We consider Dirichlet boundary conditions at one end of the spatial domain and actuation at the other end. The control design relies on the backstepping methodology and aims at assigning the distributed damping (which determines the decay rate of the solutions) of the closed-loop system. The problem is formulated using the port-Hamiltonian system framework that allows the introduction of tuning parameters with clear physical interpretations for both backstepping transformations and achievable closed-loop behavior. The overall design is carried out on the vibrating string system example. Simulations illustrate the performance of the controller.

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.

Output feedback stabilization of a first‐order hyperbolic equation with multi‐point nonlocal term

AbstractThis paper presents the output feedback stabilization of a first‐order hyperbolic equation with multi‐point nonlocal term by boundary control. First of all, we utilize the backstepping method to design a boundary state feedback controller. On the basis of equivalence of original and target system, we guarantee the exponential stability of closed‐loop system. Hereafter, the output feedback control in the light of an infinite‐dimensional observer is constructed. We demonstrate that the resulting closed‐loop system is exponentially stable via operator semigroup theory. Finally, we display the validity of the proposed controller by some numerical simulations.

Boundary Output Feedback Stabilization of the Linearized Schrödinger Equation with Nonlocal Term

In this paper, we are concerned with the boundary output feedback stabilization problem of a Schrodinger equation with a nonlocal term. Firstly, we design an explicit boundary state feedback controller by backstepping approach. Under this controller, the closed-loop system is proved to be exponentially stable from the equivalence between the original system and the target system. Then, we propose an observer-based output feedback controller by replacing the state in state feedback controller with its estimation. The resulting closed-loop system admits a unique solution which is exponentially stable. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed feedback controller.

State and output feedback boundary control of time fractional PDE–fractional ODE cascades with space‐dependent diffusivity

This study considers the boundary control problem of a time fractional partial differential equation (PDE) – fractional ordinary differential equation (ODE) cascaded system with spatially varying diffusivity and Dirichlet connection. Using the backstepping transformation, a full-state feedback law is obtained. Meanwhile, the well-posedness of kernel functions in this transformation is established theoretically. Subsequently, a Mittag-Leffler convergent boundary observer is derived, which composes with the proposed state feedback law to enable stabilisation by output feedback. With the designed state and output feedback controllers, the Mittag-Leffler stability of the closed-loop system is then proved by the fractional Lyapunov method. Finally, the results of numerical simulations are provided for fractional cascaded plants when neither kernel ODE nor kernel PDE has an explicit solution.

Boundary state feedback exponential stabilization for a one-dimensional wave equation with velocity recirculation

In this paper, we consider boundary state feedback stabilization of a one-dimensional wave equation with in-domain feedback/recirculation of an intermediate point velocity. We firstly construct an auxiliary control system which has a nonlocal term of the displacement at the same intermediate point. Then by choosing a well-known exponentially stable wave equation as its target system, we find one backstepping transformation from which a state feedback law for this auxiliary system is proposed. Finally, taking the resulting closed-loop of the auxiliary system as a new target system, we obtain another backstepping transformation from which a boundary state feedback controller for the original system is designed. By the equivalence of three systems, the closed-loop of original system is proved to be well-posed and exponentially stable. Some numerical simulations are presented to validate the theoretical results.

Boundary output feedback stabilization of transport equation with non-local term

Abstract In this paper, we are concerned with boundary output feedback stabilization of a transport equation with non-local term. First, a boundary state feedback controller is designed by a backstepping approach. The closed-loop system is proved to be exponentially stable by the equivalence between original and target system. Then, we design an output feedback controller based on an infinite-dimensional observer. It is shown that the result closed-loop system is also exponentially stable. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed feedback controller.

Event-triggered boundary feedback control for networked reaction-subdiffusion processes with input uncertainties

This paper is concerned with the event-triggered boundary feedback control problems for networked reaction-subdiffusion processes governed by time fractional reaction-diffusion systems with unknown time-varying input uncertainties over sensor/actuator networks. The event-triggered boundary state feedback controller is first designed and implemented via backstepping technique. Moreover, we realize that the availability of full-state measurements in many practical applications may be impossible due to the difficulties in measuring. To solve this limitation, we design an extended Luenberger observer that embeds within the networked sensor to estimate the whole states of the systems under consideration. Based on this, the boundary output feedback event-triggered implementation of the studied system in the context of sensor/actuator networks is then proposed. It is shown that both two kinds of event-triggered strategies could significantly asymptotically stabilize the estimation with the Zeno phenomenon being excluded. Two numerical illustrations are finally included.

Boundary output feedback control of a flexible spacecraft system with input constraint

This study is concerned with vibration control of a flexible spacecraft system in the presence of disturbances and input saturation constraint. The dynamics of the spacecraft consisting of a rigid body and two flexible appendages is presented by the coupled partial differential equations and ordinary differential equations. The first boundary state feedback control is adopted when the spacecraft system states can be measured. The second boundary output feedback control is employed when there exists the unmeasurable state in the proposed control law. A high-gain observer is developed to estimate the unmeasurable system state, two auxiliary systems are designed to prevent the presence of input saturation from destroying the system performance and two disturbance observers are established to handle the unknown boundary disturbances. The well-posedness and stability of the spacecraft system are proved. Simulation results are presented to demonstrate the effectiveness of the proposed control schemes.

Exponential stabilization of a smart piezoelectric composite beam with only one boundary controller

Layered smart composite beams involving a piezoelectric layer are traditionally actuated by a voltage source by the extension mechanism. In this paper, we consider only the bending and shear of a cantilevered piezoelectric smart composite beam modeled by the Mead-Marcus sandwich beam assumptions. Uniform exponential stabilitization with only one boundary state feedback controller, simultaneously controlling both bending moment and shear, is proved by using a spectral multiplier approach. The state feedback controller slightly differs from the classical counterparts by a non-trivial compact and nonnegative integral operator. This is due to the strong coupling of the charge equation with the stretching and bending equations. For simulations, the so-called filtered semi-discrete finite difference method is adopted.

Stabilization by forwarding design for switched feedforward systems with unstable modes

SummaryThe problem of global stabilization is investigated for a class of switched nonlinear feedforward systems in this paper where the solvability of the stabilization problem for individual subsystem is not assumed. Some sufficient condition for the stabilization problem to be solvable is derived for the first time by exploiting the multiple Lyapunov functions method and the forwarding technique. Also, we design a switching law and construct bounded state feedback controllers of subsystems explicitly by a recursive design algorithm to achieve global asymptotic stability. The provided technique permits removal of a common restriction in which all subsystems in switched nonlinear feedforward systems are globally asymptotically stable. Finally, a numerical example is provided to demonstrate the feasibility of the theoretical result. Copyright © 2017 John Wiley &amp; Sons, Ltd.