Hyperbolic Distributed Parameter Systems Research Articles

Concurrent Learning Robust Adaptive Fault Tolerant Boundary Regulation of Hyperbolic Distributed Parameter Systems.

This article develops a robust adaptive boundary output regulation approach for a class of complex anticollocated hyperbolic partial differential equations subjected to multiplicative unknown faults in both the boundary sensor and actuator. The regulator design is based on the internal model principle, which amounts to stabilize a coupled cascade system, which consists of a finite-dimensional internal model driven by a hyperbolic distributed parameter system (DPS). To this end, a systematic sliding mode equipped with a backstepping approach is developed such that the robust state feedback control can be realized. Moreover, since the available information is a faulty boundary measurement at the right side point, state estimation is required. However, due to the presence of boundary unknown faults, we need to solve an issue of joint fault-state estimation. Restrictive persistent excitation conditions are usually required to guarantee the exact estimation of faults but are unrealistic in practice. To this end, a novel concurrent learning (CL) adaptive observer is proposed so that exponential convergence is obtained. It is the first time that the spirit of CL is introduced to the field of DPSs. Consequently, the observer-based adaptive boundary fault tolerant control scheme is developed, and rigorous theoretical analysis is given such that the exponential output regulation can be achieved. Finally, the effectiveness of the proposed methodology is demonstrated via comparative simulations.

Sampling-based event-triggered iterative learning control in nonlinear hyperbolic distributed parameter systems

This article explores a sampling-based event-triggered iterative learning control (SETILC) method for achieving precise control in nonlinear hyperbolic distributed parameter systems. Firstly, by designing a Lyapunov function regarding the system output error, reasonable triggering conditions are derived to selectively update the system control input. Subsequently, a sampling-based event-triggered iterative learning control algorithm is proposed to further enhance the control performance of the system while avoiding the occurrence of Zeno phenomena. A condition sufficient for ensuring the convergence of the system’s tracking error to zero is presented, and its convergence is proven. Finally, the proposed method’s effectiveness is verified through numerical simulations.

Adaptive prescribed performance control for wave equations with dynamic boundary and multiple parametric uncertainties

<abstract><p>In modern engineering, the dynamics of many practical problems can be described by hyperbolic distributed parameter systems. This paper is devoted to the adaptive prescribed performance control for a class of typical uncertain hyperbolic distributed parameter systems, since uncertainties are inevitable in practice. The systems in question simultaneously have unknown in-domain spatially varying damping coefficient and unknown boundary constant damping coefficient. Moreover, dynamic boundary condition is considered in the present paper. These characteristics make the control problem in the paper essentially different from those in the related works. To solve the problem, using adaptive technique based projection operator, backstepping method developed for ODEs and Lyapunov stability theories, a powerful adaptive prescribed performance control scheme is proposed to successfully guarantee that all states of the resulting closed-loop system are bounded, furthermore, the original system state converges to an arbitrary prescribed small neighborhood of the origin. Compared with the existing results, the developed control schemes can not only effectively handle the serious uncertainties, but also overcome the technical difficulties in the infinite-dimensional backstepping control design method caused by the dynamic boundary condition and guarantee prescribed performance.</p></abstract>

Iterative learning control for hyperbolic distributed parameter systems based on sensor–actuator networks

AbstractThis paper proposes two kinds of iterative learning control (ILC) schemes for a class of the distributed parameter systems based on sensor–actuator networks which can be described by hyperbolic partial differential equations. A D‐type ILC algorithm is first considered and the convergent condition of the output error is obtained via the contraction mapping methodology. Then, the PD‐type ILC algorithm is considered in this hyperbolic distributed parameter systems based on sensor–actuator networks. Finally, a cable equation with air and structural damping is given to illustrate the effectiveness of the proposed methods.

System Gramians Evaluation for Hyperbolic PDEs

The paper considers the problem of gramians computation for linear hyperbolic distributed parameter systems. Two different cases are considered: vibrating string and beam systems. The presented approach is based on directly deriving the equations solutions by using time - space separation of variables and the Fourier series representation method. The initial problem framework is based on the state space formulation for infinite dimensional systems. This framework uses Riesz-spectral operators defined over Hilbert spaces and implements the concept of a C0 strongly continuous semigroup generated by bounded system operator. The solution of the hyperbolic partial differential equations is divided in two parts. The zero input part is due to the initial conditions and participates in obtaining the observability gramian of the system. The zero state part is a consequence of the input signal effect and is used to compute the controllability gramian.

Model Predictive Control for First-Order Hyperbolic System Based on Quasi-Shannon Wavelet Basis

In this paper, we consider a class of first-order hyperbolic distributed parameter systems. Our focus is on the design of a new class of model predictive control schemes using a quasi-Shannon wavelet basis. First, the first-order hyperbolic distributed parameter system is transformed into an equivalent system using collocation techniques for the approximation of spatial derivatives and Euler forward difference method for the approximation of the time component. Then, a model reduction method is applied to obtain a reduced-order system on which a nonlinear model predictive controller is designed through solving a nonlinear quadratic programming problem with input constraints. For illustration, the temperature control of a flow-control long-duct heating system is considered to be an example. A comparative simulation study is conducted to demonstrate the effectiveness of the proposed method.

Iterative learning control for boundary tracking of uncertain nonlinear wave equations

This work presents a framework of iterative learning control (ILC) design for a class of nonlinear wave equations. The main contribution lies in that it is the first time to extend the idea of well-established ILC for lumped parameter systems to boundary tracking control of nonlinear hyperbolic distributed parameter systems (DPSs). By fully utilizing the system repetitiveness, the proposed control algorithm is capable of dealing with time-space-varying and even state-dependent uncertainties. The convergence and robustness of the proposed ILC scheme are analyzed rigorously via the contraction mapping methodology and differential/integral constraints without any system dynamics simplification or discretization. In the end, two examples are provided to show the efficacy of the proposed control scheme.

Iterative learning control for MIMO second-order hyperbolic distributed parameter systems with uncertainties

In this paper, we consider an iterative learning control (ILC) problem for a class of multiinput-multioutput (MIMO) second-order hyperbolic distributed parameter systems with uncertainties. A P-type ILC scheme is proposed in the iteration procedure for distributed systems with an initial deviation in the state. A convergence of tracking error with respect to the iteration index can be guaranteed in the sense of $\mathbf{L}^{2}$ norm. Feasibility in theory of the iterative learning algorithm with difference method is proposed. Numerical simulation results are presented to illustrate the effectiveness of the proposed ILC approach.

Normal form observers for hyperbolic distributed parameter systems with spatially varying coefficients

AbstractObserver design for 1D linear distributed parameter systems of hyperbolic type with boundary measurement is discussed. The approach is based on the observer canonical form introduced on the basis of a functional‐differential equation (f.d.e.) equivalent to the original posed boundary value problem. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

ITERATIVE LEARNING CONTROL FOR SECOND ORDER NONLINEAR HYPERBOLIC DISTRIBUTED PARAMETER SYSTEMS

The problem of iterative learning control algorithm for a class of distributed parameter systems is considered.Here,the considered distributed parameter systems are composed of second order nonlinear hyperbolic partial differential equations.According to system properties,iterative learning control laws are proposed for such nonlinear hyperbolic distributed parameter systems based on P-type learning scheme.Using the contraction mapping method,it is shown that the scheme can guarantee the output tracking errors on L~2 space converge along the iteration axis.A numerical example illustrates the effectiveness of the proposed method.

Flatness based feedback design for hyperbolic distributed parameter systems with spatially varying coefficients

State feedback design for a hybrid system consisting of a spatially one-dimensional second order hyperbolic distributed parameter system with spatially dependent coefficients and a finite dimensional linear system is considered. The approach relies on the flatness based parametrization of the solutions which is obtained by integration along the characteristics. Based on this parametrization the design of flatness based feedback controllers is proposed in order to achieve an arbitrary desired closed loop dynamics. Approximation strategies of the obtained infinite dimensional control laws are proposed.

Exponential stability of non-linear hyperbolic distributed complex-valued parameter systems: The linear fuzzy operator inequality approach

In this work, exponential stability of non-linear hyperbolic distributed complex-valued parameter systems has been addressed. Using a linear fuzzy operator inequality approach, which is a novel notion proposed for the first time in this work, delay-dependent sufficient conditions for the exponential stability in complex Hilbert spaces are established in terms of linear matrix inequalities (LMIs). Finally, numerical computation illustrates our result.

A flatness based view on state controllability of simple hyperbolic distributed parameter systems. A string with interior point mass.

AbstractA flatness based approach to the analysis of state controllability of systems described by one or more boundary coupled 1D hyperbolic second order equations is analyzed on the basis of a nontrivial example. Introducing a state which corresponds to the parametrization of the solution trajectories by a so called flat output and establishing the state transformation to the original state space the controllability problem reduces to a simple interpolation problem. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A flatness based view on state controllability of simple hyperbolic distributed parameter systems. A string with interior point mass.

AbstractA flatness based approach to the analysis of state controllability of systems described by one or more boundary coupled 1D hyperbolic second order equations is analyzed on the basis of a nontrivial example. Introducing a state which corresponds to the parametrization of the solution trajectories by a so called flat output and establishing the state transformation to the original state space the controllability problem reduces to a simple interpolation problem. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Robust nonlinear H ∞ control of hyperbolic distributed parameter systems

A radial basis function (RBF) neural network model is developed for the identification of hyperbolic distributed parameter systems (DPSs). The empirical model is based only on process input–output data and is used for the estimation of the controlled variables at multiple spatial locations. The produced nonlinear model is transformed to a nonlinear state–space formulation, which in turn is used for deriving a robust H ∞ control law. The proposed methodology is applied to a long duct for the flow-based control of temperature distribution. The performance of the proposed method is illustrated by comparing it with conventional control strategies.

Feedforward Control Design for the Inviscid Burger Equation using Formal Power Series and Summation Methods

This article presents a flatness–based approach to the trajectory planning and feedforward control problem for the inviscid Burger equation with and without an additional quadratic nonlinearity. It uses the property of formal power series parameterizability of the underlying partial differential equation and uniform Euler–summability of the resulting power series to derive a parameterization of the system state and the system input in terms of a flat output. The article thereby extends the application of the formal power series approach from parabolic to first–order hyperbolic distributed–parameter systems.

Feedback control of hyperbolic distributed parameter systems

Hyperbolic distributed parameter systems (DPS) represent a large number of industrial processes with spatially nonuniform operating variable profiles. Research has been conducted to develop high-performance control strategies for these systems by exploiting their high-fidelity models. In this paper, a feedback control method that yields improved performance is proposed for DPS modelled by first-order hyperbolic partial differential equations (PDEs) using the method of characteristics. Simulation results show that this method can provide effective control for the systems modelled by a scalar PDE as well as a system of PDEs. Further, it can efficiently compensate the effect of model-plant mismatch and effectively reject the disturbances.

Duality in the Optimal Control Problems for Hyperbolic Distributed Parameter Systems with Damping Terms

In this paper, we study the duality theory of hyperbolic systems with damping terms. The main objective is to prove a duality theorem under general conditions within an infinite-dimensional framework. An example of its application is also given.

Adaptive Parameter Estimation of Hyperbolic Distributed Parameter Systems: Non-symmetric Damping and Slowly Time Varying Systems

In this paper a model reference-based adaptive parameter estimator for a wide class of hyperbolic distributed parameter systems is considered. The proposed state and parameter estimator can handle hyperbolic systems in which the damping sesquilinear form may not be symmetric (or even present) and a modification to the standard adaptive law is introduced to account for this lack of symmetry (or absence) in the damping form. In addition, the proposed scheme is modified for systems in which the input operator, bounded or unbounded, is also unknown. Parameters that are slowly time varying are also considered in this scheme via an extension of finite dimensional results. Using a Lyapunov type argument, state convergence is established and with the additional assumption of persistence of excitation, parameter convergence is shown. An approximation theory necessary for numerical implementation is established and numerical results are presented to demonstrate the applicability of the above parameter estimators.

On the persistence of excitation in the adaptive estimation of distributed parameter systems

Persistence of excitation is a sufficient condition for parameter convergence in adaptive identification schemes for dynamical systems. For abstract parabolic and hyperbolic distributed parameter systems, this condition requires that a family of bounded linear functionals be norm bounded away from zero. The level of persistence of excitation of the plant and its implications are considered for a simple parabolic and hyperbolic system. Its effect on the qualitative and quantitative behavior of the estimators is investigated. >