Showing papers by "Richard M. Murray published in 2009"

Data Transmission Over Networks for Estimation and Control

Abstract: We consider the problem of controlling a linear time invariant process when the controller is located at a location remote from where the sensor measurements are being generated. The communication from the sensor to the controller is supported by a communication network with arbitrary topology composed of analog erasure channels. Using a separation principle, we prove that the optimal linear-quadratic-Gaussian (LQG) controller consists of an LQ optimal regulator along with an estimator that estimates the state of the process across the communication network. We then determine the optimal information processing strategy that should be followed by each node in the network so that the estimator is able to compute the best possible estimate in the minimum mean squared error sense. The algorithm is optimal for any packet-dropping process and at every time step, even though it is recursive and hence requires a constant amount of memory, processing and transmission at every node in the network per time step. For the case when the packet drop processes are memoryless and independent across links, we analyze the stability properties and the performance of the closed loop system. The algorithm is an attempt to escape the viewpoint of treating a network of communication links as a single end-to-end link with the probability of successful transmission determined by some measure of the reliability of the network.

Receding horizon temporal logic planning for dynamical systems

Abstract: This paper bridges the advances in computer science and control to allow automatic synthesis of control strategies for complex dynamical systems which are guaranteed, by construction, to satisfy the desired properties even in the presence of adversary. The desired properties are expressed in the language of temporal logic. With its expressive power, a wider class of properties than safety and stability can be specified. The resulting system consists of a discrete planner that plans, in the abstracted discrete domain, a set of transitions of the system to ensure the correct behaviors and a continuous controller that continuously implements the plan. To address the computational difficulties in the synthesis of a discrete planner, we present a receding horizon based scheme for executing finite state automata that essentially reduces the synthesis problem to a set of smaller problems.

Model reduction of interconnected linear systems

Abstract: The problem of model reduction of linear systems with certain interconnection structure is considered in this paper To preserve the interconnection structure between subsystems in the reduction, special care needs to be taken This problem is important and timely because of the recent focus on complex networked systems in control engineering Two different model reduction methods are introduced and compared in this paper Both methods are extensions to the well-known balanced truncation method Compared with earlier work in the area these methods use a more general linear fractional transformation framework, and utilize linear matrix inequalities Furthermore, new approximation error bounds that reduce to classical bounds in special cases are derived The so-called structured Hankel singular values are used in the methods, and indicate how important states in the subsystems are with respect to a chosen input-output map for the entire interconnected system It is shown how these structured Hankel singular values can be used to select an approximation order Finally, the two methods are applied to a model of a mechanical device

To Drop or Not to Drop: Design Principles for Kalman Filtering Over Wireless Fading Channels

Abstract: It is the general assumption that in estimation and control over wireless links, the receiver should drop any erroneous packets. While this approach is appropriate for non real-time data-network applications, it can result in instability and loss of performance in networked control systems. In this technical note we consider estimation of a multiple-input multiple-output dynamical system over a mobile fading communication channel using a Kalman filter. We show that the communication protocols suitable for other already-existing applications like data networks may not be entirely applicable for estimation and control of a rapidly changing dynamical system. We then develop new design paradigms in terms of handling noisy packets for such delay-sensitive applications. We reformulate the estimation problem to include the impact of stochastic communication noise in the erroneous packets. We prove that, in the absence of a permanent cross-layer information path, packet drop should be designed to balance information loss and communication noise in order to optimize the performance.

Dynamics and stability of a class of low Reynolds number swimmers near a wall

Abstract: We study the dynamic stability of low Reynolds number swimming near a plane wall from a control-theoretic viewpoint. We consider a special class of swimmers having a constant shape, focus on steady motion parallel to the wall, and derive conditions under which it is passively stable without sensing or feedback. We study the geometric structure of the swimming equation and highlight the relation between stability and reversing symmetry of the dynamical system. Finally, our numerical simulations reveal the existence of stable periodic motion. The results have implications for design of miniature robotic swimmers, as well as for explaining the attraction of micro-organisms to surfaces.

On quantized consensus by means of gossip algorithm - Part I: Convergence proof

Abstract: This paper deals with the distributed averaging problem over a connected network of agents, subject to a quantization constraint. It is assumed that at each time update, only a pair of agents can update their own numbers in terms of the quantized data being exchanged. The agents are also required to communicate with one another in a stochastic fashion. In the first part of the paper, it was shown that the quantized consensus is reached by means of a stochastic gossip algorithm proposed in a recent paper, for any arbitrary quantization. The current part of the paper considers the expected value of the time at which the quantized consensus is reached. This quantity (corresponding to the worst case) is lower and upper bounded in terms of the topology of the graph, for uniform quantization. In particular, it is shown that the upper bound is related to the principal minors of the weighted Laplacian matrix. A convex optimization is also proposed to determine the set of probabilities (used to pick a pair of agents) which leads to the fast convergence of the gossip algorithm.

Compositional stability analysis based on dual decomposition

Abstract: We propose a compositional stability analysis framework for verifying properties of systems that are interconnections of multiple subsystems. The proposed method assembles stability certificates for the interconnected system based on the certificates for the input-output properties of the subsystems. The decomposition in the analysis is achieved by utilizing dual decomposition ideas from optimization. Decoupled subproblems establish subsystem level input-output properties whereas the “master” problem imposes and updates the conditions on the subproblems toward ensuring interconnected system level stability properties. Both global stability analysis and region-of-attraction analysis are discussed.

Real-valued average consensus over noisy quantized channels

Abstract: This paper concerns the average consensus problem with the constraint of quantized communication between nodes. A broad class of algorithms is analyzed, in which the transmission strategy, which decides what value to communicate to the neighbours, can include various kinds of rounding, probabilistic quantization, and bounded noise. The arbitrariness of the transmission strategy is compensated by a feedback mechanism which can be interpreted as a self-inhibitory action. The result is that the average of the nodes state is not conserved across iterations, and the nodes do not converge to a consensus; however, we show that both errors can be made as small as desired. Bounds on these quantities involve the spectral properties of the graph and can be proved by employing elementary techniques of LTI systems analysis.

Periodically Controlled Hybrid Systems

Abstract: This paper introduces Periodically Controlled Hybrid Automata (PCHA) for describing a class of hybrid control systems. In a PCHA, control actions occur roughly periodically while internal and input actions may occur in the interim changing the discrete-state or the setpoint. Based on periodicity and subtangential conditions, a new sufficient condition for verifying invariance of PCHAs is presented. This technique is used in verifying safety of the planner-controller subsystem of an autonomous ground vehicle, and in deriving geometric properties of planner generated paths that can be followed safely by the controller under environmental uncertainties.

Cone invariance and rendezvous of multiple agents

Abstract: In this article is presented a dynamical systems framework for analysing multi-agent rendezvous problems and characterize the dynamical behaviour of the collective system. Recently, the problem of rendezvous has been addressed considerably in the graph theoretic framework, which is strongly based on the communication aspects of the problem. The proposed approach is based on the set invariance theory and focusses on how to generate feedback between the vehicles, a key part of the rendezvous problem. The rendezvous problem is defined on the positions of the agents and the dynamics is modelled as linear first-order systems. These algorithms have also been applied to non-linear first-order systems. The rendezvous problem in the framework of cooperative and competitive dynamical systems is analysed that has had some remarkable applications to biological sciences. Cooperative and competitive dynamical systems are shown to generate monotone flows by the classical Muller–Kamke theorem, which is analysed using the set invariance theory. In this article, equivalence between the rendezvous problem and invariance of an appropriately defined cone is established. The problem of rendezvous is cast as a stabilization problem, with a the set of constraints on the trajectories of the agents defined on the phase plane. The n-agent rendezvous problem is formulated as an ellipsoidal cone invariance problem in the n-dimensional phase space. Theoretical results based on set invariance theory and monotone dynamical systems are developed. The necessary and sufficient conditions for rendezvous of linear systems are presented in the form of linear matrix inequalities. These conditions are also interpreted in the Lyapunov framework using multiple Lyapunov functions. Numerical examples that demonstrate application are also presented.

Kalman filtering over wireless fading channels - How to handle packet drop

Abstract: In this paper we consider estimation of dynamical systems over wireless fading communication channels using a Kalman filter. We show the impact of the stochastic communication noise on the estimation process. We furthermore show how noisy packets should be handled in the receiver. More specifically, we illustrate the impact of the availability of a cross-layer information path on the optimum receiver design. In the absence of a cross-layer information path, it was shown that packet drop should be designed to balance information loss and communication noise in order to optimize the performance. In the presence of a cross-layer path, we show that keeping all the packets will minimize the average estimation error covariance. We also derive the stability condition in the presence of noisy packets and show that it is independent of the shape of the communication noise variance or availability of a cross-layer information path.

Networked Sensing, Estimation and Control Systems

Abstract: This manuscript is for review purposes only and may not be reproduced, in whole or in part, without written consent from the author. Permission is granted for individual, non-commercial use.

A bio-plausible design for visual attitude stabilization

Abstract: We consider the problem of attitude stabilization using exclusively visual sensory input, and we look for a solution which can satisfy the constraints of a “bio-plausible” computation. We obtain a PD controller which is a bilinear form of the goal image, and the current and delayed visual input. Moreover, this controller can be learned using classic neural networks algorithms. The structure of the resulting computation, derived from general principles by imposing a bilinear computation, has striking resemblances with existing models for visual information processing in insects (Reichardt Correlators and lobula plate tangential cells). We validate the algorithms using faithful simulations of the fruit fly visual input.

A real-time helicopter testbed for insect-inspired visual flight control

Abstract: The paper describes an indoor helicopter testbed that allows implementing and testing of bio-inspired control algorithms developed from scientific studies on insects The helicopter receives and is controlled by simulated sensory inputs (eg visual stimuli) generated in a virtual 3D environment, where the connection between the physical world and the virtual world is provided by a video camera tracking system The virtual environment is specified by a 3D computer model and is relatively simple to modify compared to realistic scenes This enables rapid examinations of whether a certain control law is robust under various environments, an important feature of insect behavior As a first attempt, flight stabilization and yaw rate control near hover are demonstrated, utilizing biologically realistic visual stimuli as in the fruit fly Drosophila melanogaster

Limits on the Network Sensitivity Function for Multi-Agent Systems on a Graph

Abstract: 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.

Flight Dynamics and Control of Evasive Maneuvers: The Fruit Fly's Takeoff

Abstract: We have approached the problem of reverse-engineering the flight control mechanism of the fruit fly by studying the dynamics of the responses to a visual stimulus during takeoff. Building upon a prior framework [G. Card and M. Dickinson, J. Exp. Biol., vol. 211, pp. 341-353, 2008], we seek to understand the strategies employed by the animal to stabilize attitude and orientation during these evasive, highly dynamical maneuvers. As a first step, we consider the dynamics from a gray-box perspective: examining lumped forces produced by the insect's legs and wings. The reconstruction of the flight initiation dynamics, based on the unconstrained motion formulation for a rigid body, allows us to assess the fly's responses to a variety of initial conditions induced by its jump. Such assessment permits refinement by using a visual tracking algorithm to extract the kinematic envelope of the wings [E. I. Fontaine, F. Zabala, M. Dickinson, and J. Burdick, "Wing and body motion during flight initiation in Drosophila revealed by automated visual tracking," submitted for publication] in order to estimate lift and drag forces [F. Zabala, M. Dickinson, and R. Murray, "Control and stability of insect flight during highly dynamical maneuvers," submitted for publication], and recording actual leg-joint kinematics and using them to estimate jump forces [F. Zabala, "A bio-inspired model for directionality control of flight initiation," to be published.]. In this paper, we present the details of our approach in a comprehensive manner, including the salient results.

Design of insulating devices for in vitro synthetic circuits

Abstract: This paper describes a synthetic in vitro genetic circuit programmed to work as an insulating device. This circuit is composed of nucleic acids, which can be designed to interact according to user defined rules, and of few proteins that perform catalytic functions. A model of the circuit is derived from first principle biochemical laws. This model is shown to exhibit time-scale separation that makes its output insensitive to downstream time varying loads. Simulation results show the circuit effectiveness and represent the starting point for future experimental testing of the device.

Geometric control of particle manipulation in a two-dimensional fluid

Abstract: Manipulation of particles suspended in fluids is crucial for many applications, such as precision machining, chemical processes, bio-engineering, and self-feeding of microorganisms. In this paper, we study the problem of particle manipulation by cyclic fluid boundary excitations from a geometric-control viewpoint. We focus on the simplified problem of manipulating a single particle by generating controlled cyclic motion of a circular rigid body in a two-dimensional perfect fluid. We show that the drift in the particle location after one cyclic motion of the body can be interpreted as the geometric phase of a connection induced by the system's hydrodynamics. We then formulate the problem as a control system, and derive a geometric criterion for its nonlinear controllability. Moreover, by exploiting the geometric structure of the system, we explicitly construct a feedback-based gait that results in attraction of the particle towards the rigid body. We argue that our gait is robust and model-independent, and demonstrate it in both perfect fluid and Stokes fluid.

Probabilistic Safety Analysis of Sensor-Driven Hybrid Automata

Abstract: The control programs of complex autonomous systems that have conditional branching can be modeled as linear hybrid systems. When the state knowledge is perfect, linear hybrid systems with state- based transition conditions can be verified against a specified unsafe set using existing model checking software. This paper introduces a formal method for calculating the failure probability due to state estimation uncertainty of these sensor-driven hybrid systems. Problem complexity is described and some reduction techniques for the failure probability calculation are given. An example goal-based control program is given and the failure probability for that system is calculated.

Quantized consensus via adaptive stochastic gossip algorithm

Abstract: This paper is concerned with the distributed averaging problem over a given undirected graph. To enable every vertex to compute the average of the initial numbers sitting on the vertices of the graph, the policy is to pick an edge at random and update the values on its ending vertices based on some rules, but only in terms of the quantized data being exchanged between them. Our recent paper showed that the quantized consensus is reached under a simple updating protocol which deploys a fixed tuning factor. The current paper allows the tuning factor to be time-dependent in order to achieve two goals. First, this makes it possible to study the numerical stability of the protocol with a fixed tuning factor under a small perturbation of this parameter. Furthermore, exploiting a time-varying tuning factor facilitates the implementation of the consensus protocol and pushes the steady state of the system towards an equilibrium point, as opposed to making it oscillatory. The current paper is an important extension of our recent work, which generalizes a finite-dimensional problem to an infinite-dimensional one that is more challenging in nature.

Optimization of a Gene Oscillator using Transcriptional Time Delay

Abstract: Regulatory networks are responsible for controlling many aspects of biology including gene expression. Due to the complexity of existing networks many researchers have turned to synthetic networks designed to achieve a desired function. The Elowitz repressilator is a synthetic biological network of three repressors that exhibits oscillating gene expression, though its operation is noisy and lacks robustness in the cellular environment. This project aims to improve the quality of oscillation by decreasing frequency variability by inserting transcriptional time delay between one or more pairs of circuit elements. A test circuit containing twofluorescent proteins (CFP and YFP) behind separate, inducible promoters was constructed and used to characterize several delay fragments, with the fragment being inserted between YFP and its promoter while CFP functioned as an internal standard. A 1.5 kB non-coding (junk) DNA sequence derived from the GAL3 gene in yeast was selected because it did not appear to significantly impact expression and was expected to achieve a time delay of roughly 30 seconds. Baseline characterizations were performed on the repressilator using single-cell timelapse fluorescence microscopy with MATLAB image analysis software. The GAL3-derived fragment was inserted into the repressilator between the Lambda-CI promoter and lacI gene before the delayed repressilator was characterized similarly. Oscillatory behavior with dramatically reduced fluorescence intensity was observed having roughly the same frequency as the repressilator, but without the upward drift in baseline fluorescence.

Automatic Conversion Software for the Safety Verification of Goal-Based Control Programs

Abstract: Fault tolerance and safety verification of control systems are essential for the success of autonomous robotic systems. A control architecture called Mission Data System (MDS), developed at the Jet Propulsion Laboratory, takes a goalbased control approach. In this paper, a software algorithm for converting goal network control programs into linear hybrid systems is described. The conversion process is a bisimulation; the resulting linear hybrid system can be verified for safety in the presence of failures using existing symbolic model checkers, and thus the original goal network is verified. A moderately complex goal network control program is converted to a linear hybrid system using the automatic conversion software and then verified.

Control Program Verification for a Sample Titan Aerobot Mission

Abstract: The success of complex autonomous robotic systems depends on the quality and correctness of their fault tolerant control systems. A goal-based control approach is used to design an example control program for a proposed mission to Titan, a moon of Saturn. The state space of the example exceeds the capability of existing symbolic model checkers for hybrid automata, so a novel verication method is developed. An original design for verication software tool called SBT Checker that aids in the design of modular, state-based goal trees for the control of separate tasks in the overall mission, is discussed. The combination of the individual goal trees constitute a goal network control program that can be veried against unsafe conditions using the novel InVeriant software. InVeriant converts goal networks to hybrid systems whose special structure is exploited to quickly nd the reachable unsafe states. The Titan aerobot mission example is then veried using this new method.

Verifying A Controller for An Autonomous Vehicle

Abstract: This paper introduces Periodically Controlled Hybrid Au- tomata (PCHA) for describing a class of hybrid control systems. In a PCHA, control actions occur roughly periodically while internal and in- put actions, may occur in the interim changing the discrete-state or the setpoint. Based on periodicity and subtangential conditions, a new suf- cient condition for verifying invariance of PCHAs is presented. This technique is used in verifying safety of the planner-controller subsystem of an autonomous ground vehicle, and in deriving geometric properties of planner generated paths that can be followed safely by the controller under environmental uncertainties.