Ahmed Maidi

Boundary Control of Nonlinear Distributed Parameter Systems by Input-Output Linearization

Abstract The present study focuses on boundary control of nonlinear distributed parameter systems and deals with Dirichlet actuation. Thus, a design approach of a geometric control law that enforces stability and output tracking of a given punctual output is developed based on the notion of the characteristic index. The control performance of the proposed strategy is evaluated through numerical simulation by considering two control problems. The former concerns the control of the temperature of a thin metal rod modelled by a heat equation with a nonlinear source, and the later concerns the control of concentration of a dye in liquid medium modeled by Fick law with nonconstant diffusivity.

Open-loop optimal controller design using variational iteration method

This article presents a design approach of a finite-time open-loop optimal controller using Pontryagins minimum principle. The resulting equations constitute a two-point boundary-value problem, which is generally impossible to solve analytically and, furthermore the numerical solution is difficult to obtain due to the coupled nature of the solutions. In this paper, the variational iteration method is adopted to easily solve Hamilton equations by use of iteration formulas derived from the correction functionals corresponding to Hamilton equations. The proposed approach allows to derive the numerical solution of the optimal control problem but an analytical or approximate expression of the optimal control law can often be obtained as a function of the time variable, depending on the nature of the control problem, which is simple to implement. The different possible forms of control law that can be attained following the proposed design approach are illustrated by four application examples.

Distributed Geometric Control of Wave Equation

Abstract An approach for the geometric control of a one-dimensional non-autonomous linear wave equation is presented. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then based on geometric control concepts, a distributed control law that enforces stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modeled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation.

Distributed feedback design for systems governed by the wave equation

This article deals with the geometric control of a one-dimensional non-autonomous linear wave equation. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then, based on geometric control concepts, a distributed control law that enforces the exponential stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modelled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation by considering both tracking and disturbance rejection problems. The robustness of the controller has also been studied when the string tension is subjected to sudden variations.

Boundary geometric control of co-current heat exchanger

Abstract Abstract A control strategy is proposed to control the internal fluid temperature at the outlet of a co-current heat exchanger by manipulating the inlet external fluid temperature. The dynamic model of the heat exchanger is given by two partial differential equations. Based on nonlinear geometric control, a state-feedback law that ensures a desired performance of a measured output defined as spatial average temperature of the internal fluid is derived. Then, in order to control the outlet internal fluid temperature, a control strategy is proposed where an external controller is introduced to provide the set point of the considered measured output by taking as input the error between the outlet internal fluid temperature and its desired set point. The validity of the proposed control design and strategy is examined in simulation by considering the tracking and perturbation rejection problems.

Boundary Control of Burgers’ Equation by Input-Output Linearization

This paper addresses the boundary control law of a viscous Burgers’ equation with a Dirichlet actuation. The control objective consists in achieving a desired set point for a punctual output defined at a given spatial position. This control problem is characterized by an infinite characteristic index. To tackle this problem in the framework of geometric control, a design approach based on the concept of the characteristic index is proposed. First, it is proposed to convert the boundary control problem to an equivalent punctual control problem based both on the linearization of the equation and the Laplace transform in space domain. Then, to have a finite characteristic index, an auxiliary controlled output is defined and a control law that achieves a global linearization between an external variable (desired set point) and this auxiliary output is derived. To enforce set point tracking of the original punctual output, a control strategy is proposed based on the steady state relation between the punctual and the auxiliary outputs. The performance of the proposed control strategy is evaluated via numerical simulation runs.

Optimal Control of Nonlinear Chemical Processes Using the Variational Iteration Method

Abstract In this paper, a design approach of an optimal control of nonlinear chemical processes is proposed. The idea consists in adopting the variational iteration method to solve the non-linear Hamilton equations derived from the minimum principle theory. These equations constitute a two-point boundary value problem with a coupled nature of solutions. Thus, by considering the correction functionals of the Hamilton equations, the Lagrange multipliers are easily identified and practical iteration formulas are derived. Based on this formulas, an algorithm is developed to determine iteratively the solutions of the Hamilton equations with a desired accuracy. Once the solutions are obtained, the optimal control law can be easily deduced. The design approach is illustrated by two examples from chemical engineering.