Uncontrolled End Research Articles

Output regulation for the reaction‐diffusion equation with time‐varying disturbances and non‐collocated observation

The present article focuses on the output regulation problem for uncertain 1‐d reaction‐diffusion equation(RDE), in which the disturbance inputs are located at both the interior of the equation and uncontrolled end. Both the reference signal and the disturbances signal are specified due to an exosystem of time‐varying coefficients. The difficulty of the problem is that the control end is noncollocated with the regulated output. The so‐called feedforward control is designed by introducing an regulator, which is determined by infinite‐dimensional equation. A state observer is proposed based on the measurement output to recover the original and the external systems. The suitable feedback control has to be integrated by replacing the states in the expression of feedforward control law. As a result, exponential convergence of the error between the reference signal and the output, and the closed‐loop stability are established. Numerical results are performed the effectiveness of the results.

Adaptive output regulation for one‐dimensional parabolic equation with nonlocal term

SummaryIn this article, we develop the adaptive error feedback regulator design for one‐dimensional heat equation with a nonlocal term and disturbances which are located all the channels. First, we construct a transformation to transform the original system into the first auxiliary system, and in this system the measurable tracking error becomes output, both the disturbances located at the in‐domain and the control end are transformed to the uncontrolled end. Next, we design an observer for the first auxiliary system and an estimation mechanism for unknown parameters based on measurable tracking error. Then, by another transformation, we obtain the second auxiliary system, in which the disturbances and the reference signal are collocated with the control. Finally, the adaptive error feedback regulator is designed by using the backstepping method. The simulation results show that the regulator makes the tracking error asymptotically tend to zero and the state of the closed‐loop system is bounded.

Backstepping-based output regulation of ordinary differential equations cascaded by wave equation with in-domain anti-damping

Output regulation is considered in this paper for ordinary differential equations cascaded by a wave equation, in which both the body equations and the uncontrolled end are subject to disturbances. The disturbances are generated by an exosystem. A backstepping state-feedback regulator is first designed to force the output to track the reference signal. The design is based on solving cascaded regulator equations, and the solvability condition of the equations is characterized in terms of a transfer function and the eigenvalues of the exosystem. An observer-based output-feedback regulator is then designed to solve the output regulation problem. Finally, the regulator tracking performance is illustrated through numerical simulations.

Mechanistic modelling of infrared mediated energy transfer during the primary drying step of a continuous freeze-drying process

Conventional pharmaceutical freeze-drying is an inefficient and expensive batch-wise process, associated with several disadvantages leading to an uncontrolled end product variability. The proposed continuous alternative, based on spinning the vials during freezing and on optimal energy supply during drying, strongly increases process efficiency and improves product quality (uniformity). The heat transfer during continuous drying of the spin frozen vials is provided via non-contact infrared (IR) radiation. The energy transfer to the spin frozen vials should be optimised to maximise the drying efficiency while avoiding cake collapse. Therefore, a mechanistic model was developed which allows computing the optimal, dynamic IR heater temperature in function of the primary drying progress and which, hence, also allows predicting the primary drying endpoint based on the applied dynamic IR heater temperature. The model was validated by drying spin frozen vials containing the model formulation (3.9mL in 10R vials) according to the computed IR heater temperature profile. In total, 6 validation experiments were conducted. The primary drying endpoint was experimentally determined via in-line near-infrared (NIR) spectroscopy and compared with the endpoint predicted by the model (50min). The mean ratio of the experimental drying time to the predicted value was 0.91, indicating a good agreement between the model predictions and the experimental data. The end product had an elegant product appearance (visual inspection) and an acceptable residual moisture content (Karl Fischer).

Compensation of Wave Actuator Dynamics for Nonlinear Systems

The problem of stabilization of PDE-ODE cascades has been solved in the linear case for several PDE classes, whereas in the nonlinear case the problem has been solved only for the transport/delay PDE, namely for compensation of an arbitrary delay at the input of a nonlinear plant. Motivated by a specific engineering application in off-shore drilling, we solve the problem of stabilization of the cascade of a wave PDE with a general nonlinear ODE. Due to the presence of nonlinearities of arbitrary growth and the time-reversibility of the wave PDE, and due to the possibility of using arguments based on Lyapunov functionals or explicit solutions, several stability analysis approaches are possible. We present stability results in the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> × H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> × C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> norms for general nonlinear ODEs, as well as in the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> × L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm for linear ODEs. We specialize our general design for wave PDE-ODE cascades to the case of a wave PDE whose uncontrolled end does not drive an ODE but is instead governed by a nonlinear Robin boundary condition (a “nonlinear spring,” as in the friction law in drilling). This is the first global stabilization result for wave equations that incorporate non-collocated destabilizing nonlinearities of superlinear growth. We present two numerical examples, one with a nonlinear ODE and one with a nonlinear spring at the uncontrolled boundary of the wave PDE.

Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint

Problems with one-sided boundary controls and a homogeneous Robin boundary condition set on the uncontrolled end are considered in the class of strong generalized solutions of the variable coefficient wave equation. In the adjoint class of weak generalized solutions of the dual problems with one-sided observations, new constructive observability inequalities are obtained that differ from previously known ones by an optimal threshold time. It is shown that, in the considered functional classes, the estimated constants degenerate as the time interval length approaches the threshold. Numerical illustrations are given showing that the stability of approximate solutions to control problems can be substantially enhanced by taking into account a priori information contained in the resulting observability inequalities.

Threshold optimization in observability inequality for the wave equation with homogeneous Robin-type boundary condition

For the wave equation with variable coefficients, problems with one-side boundary controls of three basic types and a boundary condition of the third kind at the uncontrolled end are considered. For dual problems with one-side boundary observations in the classes of strong generalized solutions, new constructive observability inequalities are obtained that are superior to the earlier known ones in two respects. First, inequalities with an optimal value of the controllability-observability threshold are derived, and second, the value of the final evaluation constant is bounded away from zero on time intervals whose length is close to the critical length. This opens up a possibility of constructing stable approximate solutions to the indicated classes of dual control and observation problems on time intervals not only of an arbitrary supercritical but also of precisely critical length.

Flatness based control design for a nonlinear heavy chain model

Flatness based trajectory generation and control design for the spatially distributed nonlinear model of a planar heavy chain with freely movable suspension point is presented. By means of the method of characteristics it is shown, that the solution of the governing equations can be parametrized by the trajectory of the uncontrolled end of the chain using time-delays and advances.