Containment Control under DoS Attacks for Parabolic PDE Multi-Agent Systems

This paper considers the stabilization problem of a one-dimensional unstable heat conduction system subject to parametric variations and boundary uncertainties. This system is modeled as a parabolic partial differential equation (PDE) and is only powered from one boundary with a Dirichlet type of actuator. By taking the Volterra integral transformation, we obtain a nominal PDE with asymptotic stability characteristics in the new coordinates when an appropriate boundary control input is applied. The associated Lyapunov function can then be used for designing an infinite-dimensional sliding surface, on which the system exhibits exponential stability, invariant of the bounded matched disturbance, and is robust against certain types of parameter variations. A continuous variable structure boundary control law is employed to attain the sliding mode on the sliding surface. The proposed method can be extended to other parabolic PDE systems such as diffusion-advection system. Simulation results are demonstrated and compared with the other outstanding back-stepping control schemes.

A recently introduced class of the well-posed error systems and the corresponding robust model reference adaptive control (MRAC) laws for systems represented by parabolic or hyperbolic partial differential equations (PDEs) with spatially varying parameters is extended to encompass the disturbance rejection capability. The disturbance is modeled as a space-time-varying signal generated by a parabolic PDE with known parameters or as a spatially-varying time-invariant signal.

In this work, we explore a methodology to compute the empirical eigenfunctions for the order-reduction of nonlinear parabolic partial differential equations (PDEs) system with time-varying domain. The idea behind this method is to obtain the mapping functional, which relates the time-evolution scalar physical property solution ensemble of the nonlinear parabolic PDE with the time-varying domain to a fixed reference domain, while preserving space invariant properties of the raw solution ensemble. Subsequently, the Karhunen-Lo'eve decomposition is applied to the solution ensemble with fixed spatial domain resulting in a set of optimal eigenfunctions that capture the most energy of data. Further, the low dimensional set of empirical eigenfunctions is mapped (“pushed-back”) on the time-varying domain by an appropriate mapping resulting in the basis for the construction of the reduced-order model of the parabolic PDEs with time-varying domain. Finally, this methodology is applied in the representative cases of calculation of empirical eigenfunctions in the case of one and two dimensional model of nonlinear reaction-diffusion parabolic PDE systems with analytically defined domain evolutions. In particular, the design of both mappings which relate the raw data and function spaces transformations from the time-varying to time-invariant domain are designed to preserve dynamic features of the scalar physical property and we provide comparisons among reduced and high order fidelity models.

This paper presents an observer-based dynamic feedback control design for a linear parabolic partial differential equation (PDE) system, where a finite number of actuators and sensors are active over part thereof and at specified points in the spatial domain, respectively. In the proposed design method, a Luenberger-type PDE observer is first constructed by using the non-collocated pointwise measurements to exponentially tract the state of the PDE system. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for integrals, a sufficient condition on exponential stability of the resulting closed-loop system is presented in term of standard linear matrix inequalities (LMIs). Numerical simulation results are presented to show the effectiveness of the proposed design method.

: The prototype parabolic partial differential equation is the heat conduction equation. This paper deals with systems of parabolic equations. Such systems occur in many contexts in addition to heat conduction, e.g. biology, in nuclear reactor techniques, in economics, etc. Let the n-vector u denote the (unknown) solution of a system of n parabolic partial differential equations. An important question in the study of these systems is the boundedness of u. Many techniques and criteria have been developed to solve this problem if the system is weakly coupled, i.e. if the k sub th equation contains second order space derivatives of only u sub k, the k sub th component of u. If this is not the case, the system is said to be strongly coupled. This paper for a broad class of strongly coupled parabolic systems pointwis boundedness of the solution u is established.

We proposed an economic model predictive control (EMPC) system for parabolic partial differential equation (PDE) systems in [17]. The EMPC system assumed the knowledge of the complete state spatial profile at each sampling period. From a practical point of view, measurements of the state variables are available only at a finite number of spatial positions. To address this practical consideration, an output feedback EMPC system that accounts for both manipulated input and state constraints is developed for a quasi-linear parabolic PDE system. The EMPC system is applied to a non-isothermal tubular reactor. Two EMPC systems, each utilizing a different number of measurement sensors and formulated with various degrees of accuracy (i.e., number of modes retained from the infinite-dimensional model), are presented and compared on the basis of model accuracy, and input and state constraint satisfaction.

In this contribution the backstepping design of flatness-based tracking controllers for parabolic systems is extended by an integral action to acount for exogenous disturbances and model uncertainty. The proposed controller design only amounts to stabilizing an ODE by eigenvalue assignment and to applying the standard backstepping approach to a parabolic PDE. The robustness of the resulting disturbance rejection is demonstrated for model uncertainties, that do not destabilize the closed-loop system. Furthermore, it is shown, that the proposed design method allows the prevention of controller windup by a simple structural measure. The theoretical results are demonstrated by means of a parabolic system with spatially varying parameters.

Using a concept of strong necessary conditions we derive the Bogomolny decomposition for systems of two generalized elliptic and parabolic nonlinear partial differential equations (NPDE) of the second order. The generalization means that the equation coefficients depend on the field variables. According to the Cinquini-Cibrario criteria [18–20] the first type is characterized to be an elliptic, whereas the second one is a parabolic system. As a result we derive conditions for existence of the Bogomolny relationships.

We consider two quasi-linear initial-value Cauchy problems on ℝ d : a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·∇u, a forcing term h(t, x, u) and an initial condition u 0 ∈ L ∞(ℝ d ) ∩ C ∞(ℝ d ), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.

This paper proposes a general method for the synthesis of non-linear output feedback controllers for single-input singleoutput quasi-linear parabolic partial differential difference equation (PDDE) systems, for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Initially, a non-linear model reduction scheme which is based on combination of Galerkin's method with the concept of approximate inertial manifold is employed for the derivation of differential difference equation (DDE) systems that describe the dominant dynamics of the PDDE system. Then, these DDE systems are used as the basis for the explicit construction of non-linear output feedback controllers through combination of geometric and Lyapunov techniques. The controllers guarantee stability and enforce output tracking in the closed-loop parabolic PDDE system independently of the size of the state delay, provided that the separation of the slow and fast eigenvalues of the spatial differential operator is sufficiently large and an appropriate matrix is positive definite. The methodology is successfully employed to stabilize the temperature profile of a tubular reactor with recycle at a spatially non-uniform unstable steadystate.

Recently Zolezzi, [4] presented a necessary condition for optimality for the problem of optimal control of systems governed by parabolic partial differential equations with a first boundary condition. In this paper, we have developed (Theorem 4.3 and Corollary 4.5) a similar necessary condition for Cauchy problems (unbounded domains) that arise naturally in the optimal control problems of stochastic Ito differential systems with fixed terminal time. For the proof of our results we use several theorems of Aronson [1] which have been slightly modified to suit our requirements.

Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties

Chapter Nine - The BEM for Time Dependent Problems

A pattern-recognition method is given for identification of distributed parameter systems. This approach takes care of the system characterization which is the first and critical step in the identification problem. Systems which can be represented by parabolic partial differential equations are considered. The pattern recognition system has been designed to classify the pattern measurements into identifiable classes. The inputs to this pattern recognition system are the data concerning the inputs and observation responses of the distributed systems and the output is the classification of the distributed parameter system.

Lyapunov-based design of locally collocated controllers for semi-linear parabolic PDE systems