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<h3>Abstract</h3>
This work develops the geometry and dynamics of mechanical systems
with nonholonomic constraints and symmetry from the point of view of
Lagrangian mechanics and with a view to control theoretical
applications. The basic methodology is that of geometric mechanics
emphasizing the formulation of Lagrange d'Alembert with the use of
connections and momentum maps associated with the given symmetry
group. We begin by recalling and extending the results of Koiller from
the case of principal connections to the general Ehresmann
case. Unlike the situation with standard configuration space
constraints, the symmetry in the nonholonomic case may or may not lead
to conservation laws. In any case, the momentum map determined by the
symmetry group satisfies a useful differential equation that decouples
from the group variables. This momentum equation is shown to have the
form of a covariant derivative of the momentum equal to a component of
the internal generalized force. An alternative description using a
"body reference frame" realizes part of the momentum equation as
those components of the Euler-Poincare equations along the
symmetry directions consistent with the constraints. One of the
purposes of this paper is to derive this evolution equation for the
momentum and to distinguish geometrically and mechanically the cases
where it is conserved and those where it is not. An example of the
former is a ball or vertical disk rolling on a flat plane and an
example of the latter is the snakeboard, a modified version of the
skateboard which uses momentum coupling for locomotion generation. We
construct a synthesis of the mechanical connection and the Ehresmann
connection defining the constraints, obtaining an important new
object, the nonholonomic connection. Under conditions that include the
Chaplygin case (we use the terminology "purely kinematic") and the
case in which the momentum is conserved, it is known that one can
perform a reduction similar to Lagrangian reduction, which includes
the Routh procedure. We generalize this reduction procedure to the
case in which the nonholonomic connection is a principal connection
for the given symmetry group; this case includes all of the examples
considered in the paper and many others as well, such as the
wobblestone, the nonvertical disk and the bicycle. Another purpose of
this work is to lay the foundation for future work on mechanical
systems with control so that one can adapt well developed techniques
from holonomic systems, such as constructive controllability and
geometric phases. Although this will be the subject of future work,
the methodology of the present paper is developed with these goals in
mind.
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<p>This paper describes a technique for performing model reduction of systems with
travelling wave solutions via a Karhunen Loeve framework. +
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A cascade discrete-continuous state estimator design is presented for a class of monotone systems
with both continuous and discrete state evolution. The proposed estimator exploits the partial order preserved by
the system dynamics in order to satisfy two properties. First, its computation complexity scales with the number
of variables to be estimated instead of scaling with the size of the discrete state space. Second, a separation
principle holds: the continuous state estimation error is bounded by a monotonically decreasing function of the
discrete state estimation error, the latter one converging to zero. A multi-robot example is proposed. +
Characterizing and Prototyping Genetic Networks with Cell-Free Transcription-Translation Reactions +
A central goal of synthetic biology is to engineer cellular behavior by engineering synthetic gene networks for a variety of biotechnology and medical applications. The process of engineering gene networks often involves an iterative âdesign-build-testâ cycle, whereby the parts and connections that make up the network are built, characterized and varied until the desired network function is reached. Many advances have been made in the design and build portions of this cycle. However, the slow process of in vivo characterization of network function often limits the timescale of the testing step. Cell-free transcription-translation (TX-TL) systems offer a simple and fast alternative to performing these characterizations in cells. Here we provide an overview of a cell-free TX-TL system that utilizes the native Escherichia coli TX-TL machinery, thereby allowing a large repertoire of parts and networks to be characterized. As a way to demonstrate the utility of cell-free TX-TL, we illustrate the characterization of two genetic networks: an RNA transcriptional cascade and a protein regulated incoherent feed-forward loop. We also provide guidelines for designing TX-TL experiments to characterize new genetic networks. We end with a discussion of current and emerging applications of cell free systems. +
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A community of genetically heterogeneous cells embedded in an unmixed medium allows for sophisticated operations by retaining spatial differentiation and coordinating division-of-labor. To establish the principles of engineering reliable cell-cell communication in a heterogeneous environment, we examined how circuit parameters and spatial placement affect the range of length and time scales over which simple communication circuits interact. We constructed several "sender" and "receiver" strains with quorum-sensing signaling circuits. The sender cell colony produces acyl homoserine lactones (AHL), which diffuse across the semisolid medium. The receiver cell colony detects these signal molecules and reports by fluorescence. We have found that a single colony of one sender variant is sufficient to induce receiver response at more than 1.5cm separation. Furthermore, AHL degradase expression in receiver colonies produces a signal threshold effect and reduces the response level in subsequent receiver colonies. Finally, our investigation on the spatial placement of colonies gave rise to the design of a multicellular long-range communication array consisting of two alternating colony types. Its signal response successfully propagated colony-by-colony along a six-colony array spanning 4.8cm at a transmission velocity of 12.8 hours per colony or 0.075cm per hour. In addition, we have developed a reaction-diffusion model that recreates the observed behaviors of the many performed experiments using data-informed parameter estimates of signal diffusion, gene expression, and nutrient consumption. These results demonstrate that a mixed community of colonies can enable new patterning programs, and the corresponding model will facilitate the rational design of complex communication networks. +
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A computational approach to generate real-time, optimal trajectories for a
flight control experiment is presented. Minimum time trajectories are computed for
hover-to-hover and forward flight maneuvers. Instantaneous changes in the trajectory
constraints that model obstacles and threats are also investigated. Experimental
results using the Nonlinear Trajectory Generation software package show good closed loop
performance for both maneuvers and in the presence of obstacles. Success of
the algorithm demonstrates that high-confidence real-time trajectory generation is
achievable in spite of the highly nonlinear and non-convex nature of the problem. +
A
A computational approach to generating aggressive trajectories in real-time for
constrained mechanical systems is presented. The algorithm is based
on a combination of nonlinear control theory, spline theory, and sequential
quadratic programming. It is demonstrated that real-time trajectory
generation for constrained mechanical systems is possible by mapping the
problem to one of finding trajectory curves in a lower dimensional space.
Performance of the algorithm is compared with existing optimal
trajectory generation techniques. Numerical results are reported using the NTG
software package. +
I
A computationally efficient technique for the numerical solution of optimal control problems
is discussed. This method utilizes tools from nonlinear control theory to transform the optimal
control problem to a new, lower dimensional set of coordinates. It is hypothesized that maximizing the relative degree is directly related to minimizing the computation time. Firm evidence
of this hypothesis is given. Results are presented using the Nonlinear Trajectory Generation
(NTG) software package. +
A
A New Computational Method for Optimal Control of a Class of Constrained Systems Governed by Partial Differential Equations +
A computationally e±cient technique for the numerical solution of constrained
optimal control problems governed by one-dimensional partial differential
equations is considered in this paper. This technique utilizes inversion to map the
optimal control problem to a lower dimensional space. Results are presented using
the Nonlinear Trajectory Generation software package (NTG) showing that real-time
implementation may be possible. +
D
A cooperative control system consists of multiple, autonomous
components interacting to<br>
control their environment. Examples include air tra±c control systems,
automated factories,<br>
robot soccer teams and sensor/actuator networks. Designing such systems requires
a combination of tools from control theory and distributed systems. In this
article, we review some of these tools and then focus on the Computation and
Control Language, CCL, which we have developed as a modeling tool and a programming
language for cooperative control systems. +
F
A feedback controller closes the loop from vision to wing motion to stabilize forward flight in a simulation of Drosophila Melanogaster. +
M
A generalized model predictive control (MPC) formulation is derived that extends the existing theory to a multi-vehicle formation stabilization problem. The vehicles are individually
governed by nonlinear and constrained dynamics. The extension considers formation stabilization to a set of permissible equilibria, rather than a unique equilibrium. Simulations for three vehicle formations with input constrained dynamics on con¯guration space SE(2) are performed
using a nonlinear trajectory generation (NTG) software package developed at Caltech. Preliminary results and an outline of future work for scaling/decentralizing the MPC approach and applying it to an emerging experimental testbed are given. +
A master equation describes the continuous-time evolution of a probability distribution, and
is characterized by a simple bilinear structure and an often-high dimension. We develop a
model reduction approach in which the number of possible confiurations and corresponding
dimension is reduced, by removing improbable configurations and grouping similar ones. Error
bounds for the reduction are derived based on a minimum and maximum time scale of interest.
An analogous linear identification procedure is then presented, which computes the state and
output matrices for a predetermined configuration set. These ideas are demonstrated first in
a finite-dimensional model inspired by problems in surface evolution, and then in an infinite-
dimensional film growth master equation. +
S
A new technique for stabilizing nonholonomic systems to trajectories is presented. It is well known that such systems cannot be stabilized to a point using smooth static state feedback. In this paper we suggest the use of control laws for stabilizing a system about a trajectory, instead of a point. Given a nonlinear system and a desired (nominal) feasible trajectory, the paper gives an explicit control law which will locally exponentially stabilize the system to the desired trajectory. The theory is applied to several examples, including a car-like robot. +
A
A numerical algorithm for computing necessary conditions
for performance specifications is developed for nonlinear
uncertain systems. The algorithm is similar in nature and
behavior to the power algorithm for the mu lower bound,
and doesn't rely on a descent method. The algorithm is
applied to a practical example. +
A reactive safety mode is built into a robust model predictive control algorithm for uncertain nonlinear systems with bounded disturbances. The algorithm enforces state and control constraints and blends two modes: (I) standard, guarantees re-solvability and asymptotic convergence in a robust receding-horizon manner; (II) safety, if activated, guarantees containment within an invariant set about a reference. The reactive safety mode provides robustness to unexpected, but real-time anticipated, state-constraint changes during standard mode operation. The safety-mode control policy is designed offline and can be activated at any arbitrary time. The standard-mode control has feedforward and feedback components: feedforward is from online solution of a finite-horizon optimal control problem; feedback is designed offline to provide robustness to system uncertainty and disturbances and to establish an invariant âstate tubeâ that guarantees standard-mode re-solvability at any time. The algorithm design is shown for a class of systems with incrementally-conic uncertain/nonlinear terms and bounded disturbances. +
A reactor for the deposition of superconducting \ybcolong\;thin films is modeled and
studied from a control perspective to determine the heat transfer dynamics of the reactor
under active thermal control. A nonlinear wavelength-dependent heat transfer model is
developed to predict reactor heat transfer throughout the film growth process, and
preliminary component testing is conducted to validate the model. The model is linearized
about a typical operating point and analyzed with linear feedback control methods to
determine the performance of the reactor under observer-based feedback control with film
growth disturbances. The controller and observer are selected through linear quadratic
regulator and linear quadratic estimator methods. Rates of convergence of the controller
and observer are determined through examination of the eigenvalues of the linearized
system, and disturbance rejection is assessed with ${\mathcal H}_2$ and ${\mathcal
H}_\infty$ norms. The eigenvalue and norm analysis is applied to varying reactor design
parameters to quantify performance tradeoffs. The maximum errors associated with control
and with estimation of a nominal design case are both 21 K, and the longest time scales
are 45 seconds and 10 seconds for the controller and observer, respectively. +
L
A synthetic cell-cell signaling circuit should ideally be (1) metabolically lightweight, (2) insulated from endogenous gene networks, and (3) excitable rather than oscillatory or bistable. To accomplish these three features, we propose a synchronized pulse-generating circuit based on the design of published synchronized oscillators. This communication module employs a pulse generator built using Lux-type quorum sensing components and an IFFL transcriptional circuit. Both the input and output of this module are AHLs, the quorum sensing signaling molecule. Cells bearing this module therefore act as an excitable medium, producing a pulse of AHL when stimulated by exogenous AHL. Using simulation and microscopy, we demonstrate how this circuit enables traveling pulses of AHL production through microcolonies growing in two dimensions. Traveling pulses achieve cell-cell communication at longer distances than can be achieved by diffusion of signal from sender to receiver cells and may permit more sophisticated coordination in synthetic consortia. +
O
AbstractIn this paper, we consider the problem of
optimal Linear Quadratic Gaussian control of a system
in which communication between the sensor and the
controller occurs across a packet-dropping link. We first
prove a separation principle that allows us to solve this
problem using a standard LQR state-feedback design,
along with an optimal algorithm for propagating and using
the information across the unreliable link. Then we present
one such optimal algorithm, which consists of a Kalman
filter at the sensor side of the link, and a switched linear
filter at the controller side. Our design does not assume
any statistical model of the packet drop events, and is thus
optimal for any arbitrary packet drop pattern. Further,
the solution is appealing from a practical point of view
because it can be implemented as a small modification of
an existing LQG control design. +
