53 research outputs found
Delay-Adaptive Control of First-order Hyperbolic PIDEs
We develop a delay-adaptive controller for a class of first-order hyperbolic
partial integro-differential equations (PIDEs) with an unknown input delay. By
employing a transport PDE to represent delayed actuator states, the system is
transformed into a transport partial differential equation (PDE) with unknown
propagation speed cascaded with a PIDE. A parameter update law is designed
using a Lyapunov argument and the infinite-dimensional backstepping technique
to establish global stability results. Furthermore, the well-posedness of the
closed-loop system is analyzed. Finally, the effectiveness of the proposed
method was validated through numerical simulation
Robust stabilization of first-order hyperbolic PDEs with uncertain input delay
A backstepping-based compensator design is developed for a system of
first-order linear hyperbolic partial differential equations (PDE)
in the presence of an uncertain long input delay at boundary. We introduce a
transport PDE to represent the delayed input, which leads to three coupled
first-order hyperbolic PDEs. A novel backstepping transformation, composed of
two Volterra transformations and an affine Volterra transformation, is
introduced for the predictive control design. The resulting kernel equations
from the affine Volterra transformation are two coupled first-order PDEs and
each with two boundary conditions, which brings challenges to the
well-posedness analysis. We solve the challenge by using the method of
characteristics and the successive approximation. To analyze the sensitivity of
the closed-loop system to uncertain input delay, we introduce a neutral system
which captures the control effect resulted from the delay uncertainty. It is
proved that the proposed control is robust to small delay variations. Numerical
examples illustrate the performance of the proposed compensator
Boundary control and observation of coupled parabolic PDEs
Reaction-diffusion equations are parabolic Partial Differential Equations (PDEs) which
often occur in practice, e.g., to model the concentration of one or more substances, distributed
in space, under the in
uence of different phenomena such as local chemical reactions,
in which the substances are transformed into each other, and diffusion, which causes
the substances to spread out over a surface in space. Certainly, reaction-diffusion PDEs
are not confined to chemical applications but they also describe dynamical processes of
non-chemical nature, with examples being found in thermodynamics, biology, geology,
physics, ecology, etc.
Problems such as parabolic Partial Differential Equations (PDEs) and many others
require the user to have a considerable background in PDEs and functional analysis before
one can study the control design methods for these systems, particularly boundary control
design.
Control and observation of coupled parabolic PDEs comes in roughly two settingsdepending
on where the actuators and sensors are located \in domain" control, where
the actuation penetrates inside the domain of the PDE system or is evenly distributed
everywhere in the domain and \boundary" control, where the actuation and sensing are
applied only through the boundary conditions.
Boundary control is generally considered to be physically more realistic because actuation
and sensing are nonintrusive but is also generally considered to be the harder problem,
because the \input operator" and the "output operator" are unbounded operators.
The method that this thesis develops for control of PDEs is the so-called backstepping
control method. Backstepping is a particular approach to stabilization of dynamic
systems and is particularly successful in the area of nonlinear control. The backstepping
method achieves Lyapunov stabilization, which is often achieved by collectively shifting
all the eigenvalues in a favorable direction in the complex plane, rather than by assigning
individual eigenvalues. As the reader will soon learn, this task can be achieved in a rather
elegant way, where the control gains are easy to compute symbolically, numerically, and
in some cases even explicitly.
In addition to presenting the methods for boundary control design, we present the dual
methods for observer design using boundary sensing. Virtually every one of our control
designs for full state stabilization has an observer counterpart. The observer gains are
easy to compute symbolically or even explicitly in some cases. They are designed in
such a way that the observer error system is exponentially stabilized. As in the case of
finite-dimensional observer-based control, a separation principle holds in the sense that a
closed-loop system remains stable after a full state stabilizing feedback is replaced by a
feedback that employs the observer state instead of the plant state
Boundary control of parabolic PDE using adaptive dynamic programming
In this dissertation, novel adaptive/approximate dynamic programming (ADP) based state and output feedback control methods are presented for distributed parameter systems (DPS) which are expressed as uncertain parabolic partial differential equations (PDEs) in one and two dimensional domains. In the first step, the output feedback control design using an early lumping method is introduced after model reduction. Subsequently controllers were developed in four stages; Unlike current approaches in the literature, state and output feedback approaches were designed without utilizing model reduction for uncertain linear, coupled nonlinear and two-dimensional parabolic PDEs, respectively. In all of these techniques, the infinite horizon cost function was considered and controller design was obtained in a forward-in-time and online manner without solving the algebraic Riccati equation (ARE) or using value and policy iterations techniques.
Providing the stability analysis in the original infinite dimensional domain was a major challenge. Using Lyapunov criterion, the ultimate boundedness (UB) result was demonstrated for the regulation of closed-loop system using all the techniques developed herein. Moreover, due to distributed and large scale nature of state space, pure state feedback control design for DPS has proven to be practically obsolete. Therefore, output feedback design using limited point sensors in the domain or at boundaries are introduced. In the final two papers, the developed state feedback ADP control method was extended to regulate multi-dimensional and more complicated nonlinear parabolic PDE dynamics --Abstract, page iv
Boundary time-varying feedbacks for fixed-time stabilization of constant-parameter reaction-diffusion systems
International audienceIn this paper, the problem of fixed-time stabilization of constant-parameter reaction-diffusion partial differential equations by means of continuous boundary time-varying feedbacks is considered. Moreover, the time of convergence can be prescribed in the design. The design of time-varying feedbacks is carried out based on the backstepping approach. Using a suitable target system with a time varying-coefficient, one can state that the resulting kernel of the backstepping transformation is time-varying and rendering the control feedback to be time-varying as well. Explicit representations of the kernel solution in terms of generalized Laguerre polynomials and modified Bessel functions are derived. The fixed-time stability property is then proved. A simulation example is presented to illustrate the main results
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