Variable structure control for parabolic evolution equations

In this paper it is considered a class of infinite-dimensional control systems in a variational setting. By using a Faedo-Galerkin method, a sequence of approximating finite dimensional controlled differential equations is defined. On each of these systems a variable structure control is applied to constrain the motion on a specified surface. Under some growth assumptions the convergence of these approximations to an ideal sliding state for the infinite-dimensional system is shown. Results are then applied to the Neumann boundary control of a parabolic evolution equation.Comment: Submitted for presentation to the Joint 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005; 13 page

Sliding mode control for a nonlinear phase-field system

In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state-feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.Comment: Key words: phase field system, nonlinear boundary value problems, phase transition, sliding mode control, state-feedback control la

Synchronization of reaction–diffusion Hopfield neural networks with s-delays through sliding mode control

Synchronization of reaction–diffusion Hopfield neural networks with s-delays via sliding mode control (SMC) is investigated in this paper. To begin with, the system is studied in an abstract Hilbert space C([–r; 0];U) rather than usual Euclid space Rn. Then we prove that the state vector of the drive system synchronizes to that of the response system on the switching surface, which relies on equivalent control. Furthermore, we prove that switching surface is the sliding mode area under SMC. Moreover, SMC controller can also force with any initial state to reach the switching surface within finite time, and the approximating time estimate is given explicitly. These criteria are easy to check and have less restrictions, so they can provide solid theoretical guidance for practical design in the future. Three different novel Lyapunov–Krasovskii functionals are used in corresponding proofs. Meanwhile, some inequalities such as Young inequality, Cauchy inequality, Poincaré inequality, Hanalay inequality are applied in these proofs. Finally, an example is given to illustrate the availability of our theoretical result, and the simulation is also carried out based on Runge–Kutta–Chebyshev method through Matlab

Sliding modes for a phase-field system

In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state- feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target ~$\phi^*$. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state

Solvability and sliding mode control for the viscous Cahn-Hilliard system with a possibly singular potential

In the present contribution we study a viscous Cahn-Hilliard system where a further leading term in the expression for the chemical potential $\mu$ is present. This term consists of a subdifferential operator $S$ in $L^2(\Omega)$ (where $\Omega$ is the domain where the evolution takes place) acting on the difference of the phase variable $\varphi$ and a given state $\varphi^*$, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential $\mu$. Next, by assuming that $S=\rho\,$sign, a multiple of the sign operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for $\mu$ and under suitable conditions on $\rho$ and $\Omega$, we also prove the sliding mode property, that is, that $\varphi$ is forced to join the evolution of $\varphi^*$ in some time $T^*$ lower than the given final time $T$. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on $\varphi$.Comment: Key words: viscous Cahn-Hilliard equation, state-feedback control law, initial-boundary value problem, well-posedness, regularity, sliding mode propert

Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces

In this paper a model reduction technique is introduced for piecewise-smooth (PWS) vector fields, whose trajectories fall into a Banach space, but the domain of definition of the vector fields is a non-dense subset of the Banach space. The vector fields depend on a parameter that can assume different discrete values in two parts of the phase space and a continuous family of values on the boundary that separates the two parts of the phase space. In essence the parameter parametrizes the possible vector fields on the boundary. The problem is to find one or more values of the parameter so that the solution of the PWS system on the boundary satisfies certain requirements. In this paper we require continuous solutions. Motivated by the properties of applications, we assume that when the parameter is forced to switch between the two discrete values, trajectories become discontinuous. Discontinuous trajectories exist in systems whose domain of definition is non-dense. It is shown that under our assumptions the trajectories of such PWS systems have unique forward-time continuation when the parameter of the system switches. A finite-dimensional reduced order model is constructed, which accounts for the discontinuous trajectories. It is shown that this model retains uniqueness of solutions and other properties of the original PWS system. The model reduction technique is illustrated on a nonlinear bowed string model.Comment: 11 figures, 55 pages. Accepted for publication in Journal of Nonlinear Scienc

Fault tolerant control for nonlinear aircraft based on feedback linearization

The thesis concerns the fault tolerant flight control (FTFC) problem for nonlinear aircraft by making use of analytical redundancy. Considering initially fault-free flight, the feedback linearization theory plays an important role to provide a baseline control approach for de-coupling and stabilizing a non-linear statically unstable aircraft system. Then several reconfigurable control strategies are studied to provide further robust control performance:- A neural network (NN)-based adaption mechanism is used to develop reconfigurable FTFC performance through the combination of a concurrent updated learninglaw. - The combined feedback linearization and NN adaptor FTFC system is further improved through the use of a sliding mode control (SMC) strategy to enhance the convergence of the NN learning adaptor. - An approach to simultaneous estimation of both state and fault signals is incorporated within an active FTFC system.The faults acting independently on the three primary actuators of the nonlinear aircraft are compensated in the control system.The theoretical ideas developed in the thesis have been applied to the nonlinear Machan Unmanned Aerial Vehicle (UAV) system. The simulation results obtained from a tracking control system demonstrate the improved fault tolerant performance for all the presented control schemes, validated under various faults and disturbance scenarios.A Boeing 747 nonlinear benchmark model, developed within the framework of the GARTEUR FM-AG 16 project “fault tolerant flight control systems”,is used for the purpose of further simulation study and testing of the FTFC scheme developed by making the combined use of concurrent learning NN and SMC theory. The simulation results under the given fault scenario show a promising reconfiguration performance