35 research outputs found
Backstepping Control of Coupled General Hyperbolic-Parabolic PDE-PDE Systems
This paper considers the backstepping state feedback and observer design for
hyperbolic and parabolic PDEs, which are bidirectionally interconnected in a
general coupling structure. Both PDE subsystems consist of coupled scalar PDEs
with the heterodirectional hyperbolic PDE subsystem subject to actuation and
sensing. By making use of a multi-step approach to construct the transformation
into a stable target system, it is shown that a backstepping state feedback and
observer design only requires to solve the well-known kernel equations for the
hyperbolic and parabolic subsystems as well as additional decoupling equations.
The latter are standard initial boundary value problems for parabolic PDEs.
This significantly facilitates the well-posedness analysis and the numerical
computation of the backstepping controller. Exponential stability is verified
for the state feedback loop, the observer error dynamics, and the closed-loop
system using an observer-based compensator. The proposed backstepping design
procedures are demonstrated for numerical examples.Comment: 8 pages, 6 figures, journal paper under revie
Delay-robust stabilization of an n + m hyperbolic PDE-ODE system
International audienceIn this paper, we study the problem of stabilizing a linear ordinary differential equation through a system of an n + m (hetero-directional) coupled hyperbolic equations in the actuating path. The method relies on the use of a backstepping transform to construct a first feedback to tackle in-domain couplings present in the PDE system and then on a predictive tracking controller used to stabilize the ODE. The proposed control law is robust with respect to small delays in the control signal
Safe Adaptive Control of Hyperbolic PDE-ODE Cascades
Adaptive safe control employing conventional continuous infinite-time
adaptation requires that the initial conditions be restricted to a subset of
the safe set due to parametric uncertainty, where the safe set is shrunk in
inverse proportion to the adaptation gain. The recent regulation-triggered
adaptive control approach with batch least-squares identification (BaLSI,
pronounced ``ballsy'') completes perfect parameter identification in finite
time and offers a previously unforeseen advantage in adaptive safe control,
which we elucidate in this paper. Since the true challenge of safe control is
exhibited for CBF of a high relative degree, we undertake a safe BaLSI design
in this paper for a class of systems that possess a particularly extreme
relative degree: ODE-PDE-ODE sandwich systems. Such sandwich systems arise in
various applications, including delivery UAV with a cable-suspended load.
Collision avoidance of the payload with the surrounding environment is
required. The considered class of plants is hyperbolic PDEs
sandwiched by a strict-feedback nonlinear ODE and a linear ODE, where the
unknown coefficients, whose bounds are known and arbitrary, are associated with
the PDE in-domain coupling terms that can cause instability and with the input
signal of the distal ODE. This is the first safe adaptive control design for
PDEs, where we introduce the concept of PDE CBF whose non-negativity as well as
the ODE CBF's non-negativity are ensured with a backstepping-based safety
filter. Our safe adaptive controller is explicit and operates in the entire
original safe set
Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade Systems
This paper presents a delay-adaptive boundary control scheme for a coupled linear hyperbolic PDE-ODE cascade system with an unknown and
arbitrarily long input delay. To construct a nominal delay-compensated control
law, assuming a known input delay, a three-step backstepping design is used.
Based on the certainty equivalence principle, the nominal control action is fed
with the estimate of the unknown delay, which is generated from a batch
least-squares identifier that is updated by an event-triggering mechanism that
evaluates the growth of the norm of the system states. As a result of the
closed-loop system, the actuator and plant states can be regulated
exponentially while avoiding Zeno occurrences. A finite-time exact
identification of the unknown delay is also achieved except for the case that
all initial states of the plant are zero. As far as we know, this is the first
delay-adaptive control result for systems governed by heterodirectional
hyperbolic PDEs. The effectiveness of the proposed design is demonstrated in
the control application of a deep-sea construction vessel with cable-payload
oscillations and subject to input delay
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
Ensembles of Hyperbolic PDEs: Stabilization by Backstepping
For the quite extensively developed PDE backstepping methodology for coupled
linear hyperbolic PDEs, we provide a generalization from finite collections of
such PDEs, whose states at each location in space are vector-valued, to
previously unstudied infinite (continuum) ensembles of such hyperbolic PDEs,
whose states are function-valued. The motivation for studying such systems
comes from traffic applications (where driver and vehicle characteristics are
continuously parametrized), fluid and structural applications, and future
applications in population dynamics, including epidemiology. Our design is of
an exponentially stabilizing scalar-valued control law for a PDE system in two
independent dimensions, one spatial dimension and one ensemble dimension. In
the process of generalizing PDE backstepping from finite to infinite
collections of PDE systems, we generalize the results for PDE backstepping
kernels to the continuously parametrized Goursat-form PDEs that govern such
continuously parametrized kernels. The theory is illustrated with a simulation
example, which is selected so that the kernels are explicitly solvable, to lend
clarity and interpretability to the simulation results.Comment: 16 pages, 4 figures, to be publishe