12,467 research outputs found
Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations
The paper deals with the control and regulation by integral controllers
forthe nonlinear systems governed by scalar quasi-linear hyperbolic partial
differentialequations. Both the control input and the measured output are
located on the boundary.The closed-loop stabilization of the linearized model
with the designed integral controlleris proved first by using the method of
spectral analysis and then by the Lyapunov directmethod. Based on the
elaborated Lyapunov function we prove local exponential stabilityof the
nonlinear closed-loop system with the same controller. The output regulationto
the set-point with zero static error by the integral controller is shown
uponthe nonlinear system. Numerical simulations by the Preissmann scheme are
carriedout to validate the robustness performance of the designed controllerto
face to unknown constant disturbances
Stop-and-Go Suppression in Two-Class Congested Traffic
This paper develops boundary feedback control laws in order to damp out
traffic oscillations in the congested regime of the linearized two-class
Aw-Rascle (AR) traffic model. The macroscopic second-order two-class AR traffic
model consists of four hyperbolic partial differential equations (PDEs)
describing the dynamics of densities and velocities on freeway. The concept of
area occupancy is used to express the traffic pressure and equilibrium speed
relationship yielding a coupling between the two classes of vehicles. Each
vehicle class is characterized by its own vehicle size and driver's behavior.
The considered equilibrium profiles of the model represent evenly distributed
traffic with constant densities and velocities of both classes along the
investigated track section. After linearizing the model equations around those
equilibrium profiles, it is observed that in the congested traffic one of the
four characteristic speeds is negative, whereas the remaining three are
positive. Backstepping control design is employed to stabilize the
heterodirectional hyperbolic PDEs. The control input actuates the traffic flow
at outlet of the investigated track section and is realized by a ramp metering.
A full-state feedback is designed to achieve finite time convergence of the
density and velocity perturbations to the equilibrium at zero. This result is
then combined with an anti-collocated observer design in order to construct an
output feedback control law that damps out stop-and-go waves in finite time by
measuring the velocities and densities of both vehicle classes at the inlet of
the investigated track section. The performance of the developed controllers is
verified by simulation
Output Feedback Control of Two-lane Traffic Congestion
This paper develops output feedback boundary control to mitigate traffic
congestion of a unidirectional two-lane freeway segment. The macroscopic
traffic dynamics are described by the Aw-Rascle-Zhang (ARZ) model respectively
for both the fast and slow lanes. The traffic density and velocity of each of
the two lanes are governed by coupled nonlinear hyperbolic partial
differential equations (PDEs). Lane-changing interactions between the two lanes
lead to exchanging source terms between the two pairs second-order PDEs.
Therefore, we are dealing with nonlinear coupled hyperbolic PDEs.
Based on driver's preference for the slow and fast lanes, a reference system of
lane-specific uniform steady states in congested traffic is chosen. To
stabilize traffic densities and velocities of both lanes to the steady states,
two distinct variable speed limits (VSL) are applied at outlet boundary,
controlling the traffic velocity of each lane. Using backstepping
transformation, we map the coupled heterodirectional hyperbolic PDE system into
a cascade target system, in which traffic oscillations are damped out through
actuation of the velocities at the downstream boundary. Two full-state feedback
boundary control laws are developed. We also design a collocated boundary
observer for state estimation with sensing of densities at the outlet. Output
feedback boundary controllers are obtained by combining the collocated observer
and full-state feedback controllers. The finite time convergence to equilibrium
is achieved for both the controllers and observer designs. Numerical
simulations validate our design in two different traffic scenarios
PDE-Based Feedback Control of Freeway Traffic Flow via Time-Gap Manipulation of ACC-Equipped Vehicles
We develop a control design for stabilization of traffic flow in congested
regime, based on an Aw-Rascle-Zhang-type (ARZ-type) Partial Differential
Equation (PDE) model, for traffic consisting of both ACC-equipped (Adaptive
Cruise Control-equipped) and manual vehicles. The control input is the value of
the time-gap setting of ACC-equipped and connected vehicles, which gives rise
to a problem of control of a 2x2 nonlinear system of first-order hyperbolic
PDEs with in-domain actuation. The feedback law is designed in order to
stabilize the linearized system, around a uniform, congested equilibrium
profile. Stability of the closed-loop system under the developed control law is
shown constructing a Lyapunov functional. Convective stability is also proved
adopting an input-output approach. The performance improvement of the
closed-loop system under the proposed strategy is illustrated in simulation,
also employing four different metrics, which quantify the performance in terms
of fuel consumption, total travel time, and comfort.Comment: Submitted to IEEE Transactions on Control Systems Technology on
November 20, 2018. 8 page
Backstepping Stabilization of the Linearized Saint-Venant-Exner Model
Using the backstepping design, we achieve exponential stabilization of the
coupled Saint-Venant-Exner (SVE) PDE model of water dynamics in a
sediment-filled canal with arbitrary values of canal bottom slope, friction,
porosity, and water-sediment interaction under subcritical or supercritical
flow regime. The studied SVE model consists of two rightward convecting
transport Partial Differential Equations (PDEs) and one leftward convecting
transport PDE. A single boundary input control (with actuation located only at
downstream) strategy is adopted. A full state feedback controller is firstly
designed, which guarantees the exponential stability of the closed-loop control
system. Then, an output feedback controller is designed based on the
reconstruction of the distributed state with a backstepping observer. It also
guarantees the exponential stability of the closed-loop control system. The
flow regime depends on the dimensionless Froude number Fr, and both our
controllers can deal with the subcritical (Fr 1)
flow regime. They achieve the exponential stability results without any
restrictive conditions in contrast to existing results.Comment: 16 pages, 19 figures, Submitted to Automatic
All Mach Number Second Order Semi-Implicit Scheme for the Euler Equations of Gasdynamics
This paper presents an asymptotic preserving (AP) all Mach number finite
volume shock capturing method for the numerical solution of compressible Euler
equations of gas dynamics. Both isentropic and full Euler equations are
considered. The equations are discretized on a staggered grid. This simplifies
flux computation and guarantees a natural central discretization in the low
Mach limit, thus dramatically reducing the excessive numerical diffusion of
upwind discretizations. Furthermore, second order accuracy in space is
automatically guaranteed. For the time discretization we adopt an
Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for
the pressure in the isentropic case and for the energy in the full Euler
equations. Such equations can be solved linearly so that we do not need any
iterative solver thus reducing computational cost. Second order in time is
obtained by a suitable S-IMEX strategy taken from Boscarino et al. in [6].
Moreover, the CFL stability condition is independent of the Mach number and
depends essentially on the fluid velocity. Numerical tests are displayed in one
and two dimensions to demonstrate performances of our scheme in both
compressible and incompressible regimes
Steady-State Electrical Conduction in the Periodic Lorentz Gas
We study nonequilibrium steady states in the Lorentz gas of periodic
scatterers when an external field is applied and the particle kinetic energy is
held fixed by a ``thermostat'' constructed according to Gauss' principle of
least constraint (a model problem previously studied numerically by Moran and
Hoover). The resulting dynamics is reversible and deterministic, but does not
preserve Liouville measure. For a sufficiently small field, we prove the
following results: (1) existence of a unique stationary, ergodic measure
obtained by forward evolution of initial absolutely continuous distributions,
for which the Pesin entropy formula and Young's expression for the fractal
dimension are valid; (2) exact identity of the steady-state thermodyamic
entropy production, the asymptotic decay of the Gibbs entropy for the
time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an
explicit expression for the full nonlinear current response (Kawasaki formula);
and (4) validity of linear response theory and Ohm's transport law, including
the Einstein relation between conductivity and diffusion matrices. Results (2)
and (4) yield also a direct relation between Lyapunov exponents and zero-field
transport (=diffusion) coefficients. Although we restrict ourselves here to
dimension the results carry over to higher dimensions and to some other
physical situations: e.g. with additional external magnetic fields. The proofs
use a well-developed theory of small perturbations of hyperbolic dynamical
systems and the method of Markov sieves, an approximation of Markov partitions.
In our context we discuss also the van Kampen objection to linear response
theory, which, we point out, overlooks the ``structural stability'' of strongly
hyperbolic flows.Comment: CYCLER Paper 93feb00
Bilateral Boundary Control of Moving Shockwave in LWR Model of Congested Traffic
We develop backstepping state feedback control to stabilize a moving
shockwave in a freeway segment under bilateral boundary actuations of traffic
flow. A moving shockwave, consisting of light traffic upstream of the shockwave
and heavy traffic downstream, is usually caused by changes of local road
situations. The density discontinuity travels upstream and drivers caught in
the shockwave experience transitions from free to congested traffic. Boundary
control design in this paper brings the moving shockwave front to a static
setpoint position, hindering the upstream propagation of traffic congestion.
The traffic dynamics are described with Lighthill-Whitham-Richard (LWR) model,
leading to a system of two first-order hyperbolic partial differential
equations (PDEs). Each represents the traffic density of a spatial domain
segregated by the moving interface. By Rankine-Hugoniot condition, the
interface position is driven by flux discontinuity and thus governed by a PDE
state dependent ordinary differential equation (ODE). For the PDE-ODE coupled
system. the control objective is to stabilize both the PDE states of traffic
density and the ODE state of moving shock position to setpoint values. Using
delay representation and backstepping method, we design predictor feedback
controllers to cooperatively compensate state-dependent input delays to the
ODE. From Lyapunov stability analysis, we show local stability of the
closed-loop system in norm. The performance of controllers is
demonstrated by numerical simulation
Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems
The paper deals with output feedback stabilization of exponentially stable
systems by an integral controller. We propose appropriate Lyapunov functionals
to prove exponential stability of the closed-loop system. An example of
parabolic PDE (partial differential equation) systems and an example of
hyperbolic systems are worked out to show how exponentially stabilizing
integral controllers are designed. The proof is based on a novel Lyapunov
functional construction which employs the forwarding techniques
Reinforcement Learning versus PDE Backstepping and PI Control for Congested Freeway Traffic
We develop reinforcement learning (RL) boundary controllers to mitigate
stop-and-go traffic congestion on a freeway segment. The traffic dynamics of
the freeway segment are governed by a macroscopic Aw-Rascle-Zhang (ARZ) model,
consisting of quasi-linear partial differential equations (PDEs)
for traffic density and velocity. Boundary stabilization of the linearized ARZ
PDE model has been solved by PDE backstepping, guaranteeing spatial norm
regulation of the traffic state to uniform density and velocity and ensuring
that traffic oscillations are suppressed. Collocated Proportional (P) and
Proportional-Integral (PI) controllers also provide stability guarantees under
certain restricted conditions, and are always applicable as model-free control
options through gain tuning by trail and error, or by model-free optimization.
Although these approaches are mathematically elegant, the stabilization result
only holds locally and is usually affected by the change of model parameters.
Therefore, we reformulate the PDE boundary control problem as a RL problem that
pursues stabilization without knowing the system dynamics, simply by observing
the state values. The proximal policy optimization, a neural network-based
policy gradient algorithm, is employed to obtain RL controllers by interacting
with a numerical simulator of the ARZ PDE. Being stabilization-inspired, the RL
state-feedback boundary controllers are compared and evaluated against the
rigorously stabilizing controllers in two cases: (i) in a system with perfect
knowledge of the traffic flow dynamics, and then (ii) in one with only partial
knowledge. We obtain RL controllers that nearly recover the performance of the
backstepping, P, and PI controllers with perfect knowledge and outperform them
in some cases with partial knowledge
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