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A
Analysis of Delays in Transcriptional Signaling Networks with Time-Varying Temperature-Dependent Rate Coefficients +
This paper provides preliminary work in an aim to fundamentally understand the effects of temperature fluctuations in the dynamics of biological oscillators. Motivated by circadian rhythms, we are interested in understanding how time-varying temperatures might play a role in the properties of biochemical oscillators. This paper investigates time-dependent Arrhenius scaling of biochemical networks with delays. We assume these time-delays arise from a sequence of simpler reactions that can be modeled as an aggregate delay. We focus on a model system, the Goodwin oscillator, in which we use time-varying rate coefficients as a mechanism to understand the possible effects of temperature fluctuations. The emergence of delays from a sequence of reactions can be better understood through the Goodwin model. For a high order system and comparably high reaction rates, one can approximate the large sequence of reactions in the model with a delay, which can be interpreted as the time needed to go through the âqueueâ. Such types of delays can arise in the process of transcription for example. To study how these delays are affected by temperature fluctuations, we take the limit as the order of the system and the mean reaction rates approach infinity with a periodically time-varying rate coefficient. We show that the limit cycle of the Goodwin oscillator varies only in the limit when the oscillator frequency is much larger than the frequency of temperature oscillations. Otherwise, the instantaneous frequency of the oscillator is dominated by the mean value of the time-varying temperature. +
E
This paper studies the evaluation of learning- based object detection models in conjunction with model-checking of formal specifications defined on an abstract model of an autonomous system and its environment. In particular, we define two metrics – proposition-labeled and class-labeled confusion matrices – for evaluating object detection, and we incorporate these metrics to compute the satisfaction probability of system-level safety requirements. While confusion matrices have been effective for comparative evaluation of classification and object detection models, our framework fills two key gaps. First, we relate the performance of object detection to formal requirements defined over downstream high-level planning tasks. In particular, we provide empirical results that show that the choice of a good object detection algorithm, with respect to formal requirements on the overall system, significantly depends on the downstream planning and control design. Secondly, unlike the traditional confusion matrix, our metrics account for variations in performance with respect to the distance between the ego and the object being detected. We demonstrate this framework on a car-pedestrian example by computing the satisfaction probabilities for safety requirements formalized in Linear Temporal Logic (LTL). +
F
This paper summarizes the findings and recommendations of a recent panel on
Future Directions in Control, Dynamics, and Systems, sponsored by the US Air
Force Office of Scientific Research. A set of grand challenges that
illustrate some of the recommendations and opportunities are provided.
Finally, the paper describes two new courses being developed at Caltech that
are aligned with the recommendations of the report. +
S
This report describes Team Caltech's technical approach and results for the 2007 DARPA Urban Challenge. Our primary technical thrusts were in three areas: (1) mission and contingency management for autonomous systems; (2) distributed sensor fusion, mapping and situational awareness; and (3) optimization-based guidance, navigation and control. Our autonomous vehicle, Alice, demonstrated new capabiliites in each of these areas and drove approximately 300 autonomous miles in preparation for the race. The vehicle completed 2 of the 3 qualification tests, but did not ultimately qualify for the race due to poor performance in the merging tests at the National Qualifying
Event. +
L
This report explores the tradeoffs and limits of performance in feedback control of interconnected multi-agent systems, focused on the network sensitivity functions. We consider the interaction topology described by a directed graph and we prove that the sensitivity transfer functions between every pair of agents, arbitrarily connected, can be derived using a version of the Mason's Direct Rule. Explicit forms for special types of graphs are presented. An analysis of the role of cycles points out that these structures influence and limit considerably the performance of the system. The more the cycles are equally distributed among the formation, the better performance the system can achieve, but they are always worse than the single agent case. We also prove the networked version of Bode's integral formula, showing that it still holds for multi-agent systems. +
A
This report presents techniques for using discrete finite
rotations to reorient a spacecraft from a given initial attitude to a
final attitude which satisfies a specified aiming objective. The
objective may be a fully specified final orientation or it may require
that the spacecraft direct an instrument along a certain
direction. Constraints are also imposed on the allowable intermediate
orientations that the spacecraft may assume during the course of the
maneuver, representing the operational requirements of onboard
instrumentation. The algorithms presented consider solutions that
will achieve the desired objective with only one or two slew
maneuvers, although they may be easily extended to consider more
complicated solutions requiring additional maneuvers. +
O
This research addresses the path planning problem
with a nonlinear optimization method running in real
time. An optimization problem is continually solved to find a
time-optimal, dynamically feasible trajectory from the vehicleâs
position to some receding horizon ahead (20m-70m forward).
The locally optimal numerical solver optimizes both the spatial
and temporal components of the trajectory simultaneously, and
feeds its output to a trajectory-following controller. The method
has been implemented and tested on a modified Ford E350
van. Using one stereo pair and four LADAR units as terrain
sensors, the vehicle was able to consistently traverse a 2 mile
obstacle course at the DGC qualifying event. At the main
DGC event, the vehicle drove 8 autonomous miles through the
Nevada desert before experiencing non-planning issues. During
this time, the planning system generated a plan 4.28 times
per second on average. This execution speed, coupled with a
feedback-based trajectory-following controller was shown to be
adequate at providing smooth and reliable obstacle avoidance
even on complicated terrain. +
C
Control Theory for Synthetic Biology: Recent Advances in System Characterization, Control Design, and Controller Implementation for Synthetic Biology +
This survey aims to provide a general overview of relevant terms and resources for understanding the intersection of synthetic biology and control theory. A reader with a background in control theory should come away with a reasonable understanding of the current 24 state-of-the-art of biological system identification, controller design and implementation, and the open challenges facing the field. Additionally, this review updates and builds upon previous publications on this subject. As this particular work is limited to a selected number of topics, additional reviews are suggested throughout the text for deeper reading. In the following sections, each of the challenges is addressed within the typical workflow for control implementation of more traditionally engineered systems (Figure 1). Engineered biological systems present a number of challenges to all stages of this workflow for reasons such as limitations in real-time measurement, resource competition with the host organism, and incomplete knowledge of underlying biological processes. First, strategies for framing a biologi- cal organism as a system with defined inputs, outputs, sensors, actuators, and measurements are discussed (Figure 1a). Obtaining dynamic and reliable measurements within biological organisms is a daunting challenge, engineered or otherwise. An overview of the state-of-the-art tools for modeling and characterizing biological systems is presented, followed by system identification methods specifically designed for the types of data available from biological measurements. The difficulty in engineering complex genetic networks, combined with severe limitations in real- time measurement, means that the body of work for controller design (Figure 1b) is limited – as a result, we discuss the open problems and challenges awaiting the entrepreneurial reader, and also present a number of examples of feedback loop implementation in living cells (Figure 1c). Finally, the necessary challenges in synthetic biology and development of control theoretical frameworks that need to be addressed in order to advance the field are discussed.
S
This thesis addresses the problem of estimating the state in multi-agent decision and
control systems. In particular, a novel approach to state estimation is developed that uses
partial order theory in order to overcome some of the severe computational complexity
issues arising in multi-agent systems. Within this approach, state estimation algorithms are
developed, which enjoy proved convergence properties and are scalable with the number
of agents.
<p>
The dynamic evolution of the systems under study are characterized by the interplay of
continuous and discrete variables. Continuous variables usually represent physical quan-
tities such as position, velocity, voltage, and current, while the discrete variables usually
represent quantities internal to the decision protocol that is used for coordination, com-
munication, and control. Within the proposed state estimation approach, the estimation of
continuous and discrete variables is developed in the same mathematical framework, as a
joint continuous-discrete space is considered for the estimator. This way, the dichotomy
between the continuous and discrete world is overcome for the purpose of state estimation.
<p>
Application examples are considered, which include the state estimation in competi-
tive multi-robot systems and in multi-agent discrete event systems, and the monitoring of
distributed environments. +
T
This thesis explores the paradigm of two degree of freedom design for
nonlinear control systems. In two degree of freedom design one
generates an explicit trajectory for state and input around which the
system is linearized. Linear techniques are then used to stabilize the
system around the nominal trajectory and to deal with
uncertainty. This approach allows the use of the wealth of tools in
linear control theory to stabilize a system in the face of
uncertainty, while exploiting the nonlinearities to increase
performance. Indeed, this thesis shows through simulations and
experiments that the generation of a nominal trajectory allows more
aggressive tracking in mechanical systems.
<p>
The generation of trajectories for general systems involves the
solution of two point boundary value problems which are hard to solve
numerically. For the special class of differentially flat systems
there exists a unique correspondence between trajectories in the
output space and the full state and input space. This allows us to
generate trajectories in the lower dimensional output space where we
don't have differential constraints, and subsequently map these to the
full state and input space through an algebraic procedure. No
differential equations have to be solved in this process. This thesis
gives a definition of differential flatness in terms of differential
geometry, and proves some properties of flat systems. In particular,
it is shown that differential flatness is equivalent to dynamic
feedback linearizability in an open and dense set.
<p>
This dissertation focuses on differentially flat systems. We describe
some interesting trajectory generation problems for these systems, and
present software to solve them. We also present algorithms and
software for real time trajectory generation, that allow a
computational tradeoff between stability and performance. We prove
convergence for a rather general class of desired trajectories. If a
system is not differentially flat we can approximate it with a
differentially flat system, and extend the techniques for flat
systems. The various extensions for approximately flat systems are
validated in simulation and experiments on a thrust vectored
aircraft. A system may exhibit a two layer structure where the outer
layer is a flat system, and the inner system is not. We call this
structure \emph{outer flatness}. We investigate trajectory generation
for these systems and present theorems on the type of tracking we can
achieve. We validate the outer flatness approach on a model helicopter
in simulations and experiment.
N
This thesis focuses on understanding the use of air injection as a
means of controlling rotating stall in an axial flow compressor,
involving modeling, dynamical systems analysis, and experimental
investigations.
<p>
The first step towards this understanding was the development
of a low order model for air injection control, the starting point of
which was the Moore and Greitzer model for axial flow compressors. The
Moore and Greitzer model was extended to include the effects of air
injection and bifurcation analysis was performed to determine how the closed
loop system dynamics are different from those of the open loop system.
This low order model was then used to determine the optimal placement of
the air injection actuators.
<p>
Experimental work focused on verifying that the low order model,
developed for air injection actuation, qualitatively captured the
behavior of the Caltech compressor rig. Open loop tests were performed
to determine how the placement of the air injectors on the rig
affected the performance of the compressor. The positioning of the air
injectors that provided the greatest control authority were used in
the development of air injection controllers for rotating stall. The
controllers resulted in complete elimination of the hysteresis
associated with rotating stall. The use of a throttle actuator for the
control of the surge dynamics was investigated, and then combined with
an air injection controller for rotating stall; the resulting
controller performed quite well in throttle disturbance rejection
tests.
<p>
A higher order model was developed to qualitatively match the
experimental results with a simulation. The results of this modeling
effort compared quite well with the experimental results for the open
loop behavior of the Caltech rig. The details of how the air injection
actuators affect the compressor flow were included in this model, and
the simulation predicted the same optimal controller that was
developed through experimentation.
<p>
The development of the higher order model also included the
investigation of systematic methods for determining the simulation parameters.
Based on experimental measurements of compression system transients, the
open loop simulation parameters were identified, including values for
the compressor performance characteristic in regions where direct
measurements were not possible. These methods also provided information on
parameters used in the modeling of the pressure rise delivered by the
compressor under unsteady flow conditions.
<p>
C
This tutorial describes TuLiP, the Temporal Logic Planning toolbox, a collection of tools for designing controllers
for hybrid systems from specifications in temporal logic. The
tools support a workflow that starts from a description of
desired behavior, and of the system to be controlled. The
system can have discrete state, or be a hybrid dynamical
system with a mixed discrete and continuous state space. The
desired behavior can be represented with temporal logic and
discrete transition systems. The system description can include
uncontrollable variables that take discrete or continuous values,
and represent disturbances and other environmental factors
that affect the dynamics, as well as communication signals that
affect controller decisions.
A control design problem is solved in phases that involve
abstraction, discrete synthesis, and continuous feedback control.
Abstraction yields a discrete description of system dynamics in
logic. For piecewise affine dynamical systems, this abstraction
is constructed automatically, guided by the geometry of the dynamics
and under logical constraints from the specification. The
resulting logic formulae describe admissible discrete behaviors
that capture both controlled and environment variables. The
discrete description resulting from abstraction is then conjoined
with the desired logic specification. To find a controller, the
toolbox solves a game of infinite duration. Existence of a discrete
(winning) strategy for the controlled variables in this game is a
proof certificate for the existence of a controller for the original
problem, which guarantees satisfaction of the specification. This
discrete strategy, concretized by using continuous controllers,
yields a feedback controller for the original hybrid system. The
toolbox frontend is written in Python, with backends in C,
Python, and Cython.
The tutorial starts with an overview of the theory behind
TuLiP, and of its software architecture, organized into specifi-
cation frontends and backends that implement algorithms for
abstraction, solving games, and interfaces to other tools. Then,
the main elements for writing a specification for input to TuLiP
are introduced. These include logic formulae, discrete transition
systems annotated with predicates, and hybrid dynamical systems,
with linear or piecewise affine continuous dynamics. The
working principles of the algorithms for predicate abstraction
and discrete game solving using nested fixpoints are explained,
by following the input specification through the various transformations
that compile it to a symbolic representation that
scales well to solving large games. The tutorial concludes
with several design examples that demonstrate the toolbox’s
capabilities.
N
This work develops the geometry and dynamics of mechanical systems with nonholonomic
constraints and symmetry from the perspective of Lagrangian mechanics and with a view to
control theoretical applications. The basic methodology is that of geometric mechanics
applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and
momentum maps associated with a given symmetry group to this case. We begin by formulating
the mechanics of nonholonomic systems using an Ehresmann connection to model the
constraints, and show how the curvature of this connection enters into Lagrange's
equations. Unlike the situation with standard configuration space constraints, the
presence of symmetries in the nonholonomic case may or may not lead to conservation laws.
However, the momentum map determined by the symmetry group still satisfies a useful
differential equation that decouples from the group variables. This momentum equation,
which plays an important role in control problems, involves parallel transport operators
and is computed explicitly in coordinates. An alternative description using a ``body
reference frame'' relates part of the momentum equation to the components of the
Euler-Poincar\'{e} equations along those 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 we call the nonholonomic connection. When
the nonholonomic connection is a principal connection for the given symmetry group, we
show how to perform Lagrangian reduction in the presence of nonholonomic constraints,
generalizing previous results which only held in special cases. Several detailed examples
are given to illustrate the theory.
M
This work discusses feasibility aspects of motion planning for
groups of agents connected by a range-constrained wireless
network. Specifically, we address the difficulties encountered
when trajectories are required to preserve the connectedness of
the network. The analysis utilizes a quantity called the
connectivity robustness of the network, which can be
calculated in a distributed fashion, and thus is applicable to
distributed motion planning problems arising in control of vehicle
networks. Further, these results show that network constraints
posed as connectivity robustness constraints have minimal impact
on reachability, provided that an appropriate topology control
algorithm is implemented. This contrasts with more naive
approaches to connectivity maintenance, which can significantly
reduce the reachable set. +
D
This work examines several dynamical aspects of average consensus
in mobile networks. The results herein allow consensus on general
time-varying signals, and allow tracking analysis using standard
frequency-domain techniques. Further, the frequency-domain
analysis naturally inspires a robust small-gain version of the
algorithm, which tolerates arbitrary non-uniform time delays.
Finally, we show how to exploit a dynamical conservation property
in order to ensure consensus tracking despite splitting and
merging of the underlying mobile network. +
T
This work illuminates a theory of locomotion rooted in geometric
mechanics and nonlinear control. We regard the internal configuration
of a deformable body, together with its position and orientation in
ambient space, as a point in a trivial principal fiber bundle over the
manifold of body deformations. We obtain connections on such bundles
which describe the nonholonomic constraints, conservation laws, and
force balances to which certain propulsors are subject, and construct
and analyze control-affine normal forms for different classes of
systems. We examine the applicability of results involving geometric
phases to the practical computation of trajectories for systems
described by single connections. We propose a model for planar
carangiform swimming based on reduced Euler-Lagrange equations for the
interaction of a rigid body and an incompressible fluid, accounting
for the generation of thrust due to vortex shedding through controlled
coupling terms. We investigate the correct form of this coupling
experimentally with a robotic propulsor, comparing its observed
behavior with that predicted numerically. +
D
This work is an extension to a companion paper describing
consensus-tracking for networked agents, and shows how those
results can be applied to obtain least-squares fused estimates
based on spatially distributed measurements. This mechanism is
very robust to changes in the underlying network topology and
performance, making it an interesting candidate for sensor fusion
on autonomous mobile networks. We conclude with an example of a
preliminary application to distributed Kalman Filtering using the
proposed technique, illustrating the dependence of the performance
on the structure of the underlying network. +
C
Computing Augmented Finite Transition Systems to Synthesize Switching Protocols for Polynomial Switched Systems +
This work is motivated by the problem of synthe- sizing mode sequences for continuous-time polynomial switched systems in order to guarantee that the trajectories of the system satisfy certain high-level specifications expressed in linear temporal logic. We use augmented finite transition systems as abstract models of continuous switched systems. Augmented finite transition systems are equipped with liveness properties that can be used to enforce progress in accordance with the underlying dynamics. We then introduce abstraction and refinement relations that induce a preorder on this class of finite transition systems. By construction, the resulting pre-order respects the feasibility (i.e., realizability) of the synthesis problem. Hence, existence of a discrete switching strategy for one of these abstract finite transition systems guarantees the existence of a mode sequence for the continuous system such that all of its trajectories satisfy the specification. We also present an algorithm, which can be implemented using sum-of-squares based relaxations, to compute such high fidelity abstract models in a computationally tractable way. Finally, these ideas are illustrated on an example. +
S
This work proposes a language for describing reactive synthesis problems that integrates imperative and declarative elements. The semantics is defined in terms of two-player turn-based infinite games with full information. Currently, synthesis tools accept linear temporal logic (LTL) as input, but this description is less structured and does not facilitate the expression of sequential constraints. This motivates the use of a structured programming language to specify synthesis problems. Transition systems and guarded commands serve as imperative constructs, expressed in a syntax based on that of the modeling language Promela. The syntax allows defining which player controls data and control flow, and separating a program into assumptions and guarantees. These notions are necessary for input to game solvers. The integration of imperative and declarative paradigms allows using the paradigm that is most appropriate for expressing each requirement. The declarative part is expressed in the LTL fragment of generalized reactivity(1), which admits efficient synthesis algorithms. The implementation translates Promela to input for the Slugs synthesizer and is written in Python. +
This work proposes a symbolic algorithm for the construction of assume-guarantee specifications that allow multiple agents to coop- erate. Each agent is assigned goals expressed in a fragment of linear temporal logic known as gener- alized Streett with one pair, GR(1). These goals may be unrealizable, unless each agent makes additional assumptions, about the behavior of other agents. The algorithm constructs a con- tract among the agents, in that only the infinite behavior of the given goals is constrained, known as liveness, not the finite one, known as safety. This defers synthesis to a later stage of refinement, modularizing the design process. We prove that there exist GR(1) games that do not admit any GR(1) contract. For this reason, we formulate contracts with nested GR(1) properties and auxiliary communication variables, and prove that they always exist. The algorithmâs fixpoint structure is similar to GR(1) synthesis, enjoying time complexity polynomial in the number of states, and linear in number of recurrence goals. +
