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Probabilistic Reachability for Stochastic Hybrid Probabilistic Reachability for Stochastic Hybrid Systems: Theory, Computations, and Applications by Alessandro Abate Laurea (Universit`a

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  • Probabilistic Reachability for Stochastic Hybrid Systems: Theory, Computations, and Applications

    by

    Alessandro Abate

    Laurea (Università degli Studi di Padova) 2002 M.S. (University of California, Berkeley) 2004

    A dissertation submitted in partial satisfaction of the requirements for the degree of

    Doctor of Philosophy

    in

    Electrical Engineering and Computer Sciences

    in the

    GRADUATE DIVISION

    of the

    UNIVERSITY OF CALIFORNIA, BERKELEY

    Committee in charge:

    Professor Shankar S. Sastry, Chair Professor Pravin Varaiya Professor Claire J. Tomlin Professor Steven N. Evans

    Fall 2007

  • The dissertation of Alessandro Abate is approved:

    Professor Shankar S. Sastry, Chair Date

    Professor Pravin Varaiya Date

    Professor Claire J. Tomlin Date

    Professor Steven N. Evans Date

    University of California, Berkeley

    Fall 2007

  • Probabilistic Reachability for Stochastic Hybrid Systems:

    Theory, Computations, and Applications

    Copyright c© 2007

    by

    Alessandro Abate

  • Abstract

    Probabilistic Reachability for Stochastic Hybrid Systems:

    Theory, Computations, and Applications

    by

    Alessandro Abate

    Doctor of Philosophy in Electrical Engineering and Computer Sciences

    University of California, Berkeley

    Professor Shankar S. Sastry, Chair

    Stochastic Hybrid Systems are probabilistic models suitable at describing the

    dynamics of variables presenting interleaved and interacting continuous and discrete

    components.

    Engineering systems like communication networks or automotive and air traffic

    control systems, financial and industrial processes like market and manufacturing

    models, and natural systems like biological and ecological environments exhibit com-

    pound behaviors arising from the compositions and interactions between their hetero-

    geneous components. Hybrid Systems are mathematical models that are by definition

    suitable to describe such complex systems. The effect of the uncertainty upon the

    involved discrete and continuous dynamics—both endogenously and exogenously to

    the system—is virtually unquestionable for biological systems and often inevitable

    for engineering systems, and naturally leads to the employment of stochastic hybrid

    models.

    The first part of this dissertation introduces gradually the modeling framework

    and focuses on some of its features. In particular, two sequential approximation

    procedures are introduced, which translate a general stochastic hybrid framework

    into a new probabilistic model. Their convergence properties are sketched. It is

    argued that the obtained model is more predisposed to analysis and computations.

    The kernel of the thesis concentrates on understanding the theoretical and com-

    putational issues associated with an original notion of probabilistic reachability for

    1

  • controlled stochastic hybrid systems. The formal approach is based on formulating

    reachability analysis as a stochastic optimal control problem, which is solved via dy-

    namic programming. A number of related and significant control problems, such as

    that of probabilistic safety, are reinterpreted with this approach. The technique is

    also computationally tested on a benchmark case study throughout the whole work.

    Moreover, a methodological application of the concept in the area of Systems Biology

    is presented: a model for the production of antibiotic as a component of the stress

    response network for the bacterium Bacillus subtilis is described. The model allows

    one to reinterpret the survival analysis for the single bacterial cell as a probabilistic

    safety specification problem, which is then studied by the aforementioned technique.

    In conclusion, this dissertation aims at introducing a novel concept of probabilis-

    tic reachability that is both formally rigorous, computationally analyzable and of

    applicative interest. Furthermore, by the introduction of convergent approximation

    procedures, the thesis relates and positively compares the presented approach with

    other techniques in the literature.

    Professor Shankar S. Sastry, Chair Date

    2

  • Acknowledgements

    This dissertation is the final outcome of the work of a single person, but benefits from the contributions of many. I would like to acknowledge the help I have had in the past few years of graduate school at Berkeley. Shankar Sastry, my advisor, has granted me the unconditional intellectual freedom to pursue my ideas and intuitions, the enthusiasm to aim higher, and the financial tranquillity that enabled me to focus on my work. Maria Prandini and John Lygeros have contributed to this project in first person, dispensing their attention and their knowledge of Hybrid Systems on me. I greatly benefited from John’s expertize and Maria’s technical rigor. Claire Tomlin has always been ready to give me extremely effective feedback on my work, and I am excited about our upcoming collaboration. Steven Evans, in the department of Statistics, and Pravin Varaiya in EECS have al- ways been ready to help me and gave me competent and thorough feedback as members of my committee. I have learned a lot from my collaboration with the CS Lab at SRI International, in particular with Ashish Tiwari. Laurent El Ghaoui and Slobodan Simic have been reference figures for me, as well as collaborators, especially during my Master years. Indirectly, I have had the chance to benefit from the teaching and the exchange of ideas with exceptional people at Berkeley: Alberto Sangiovanni-Vincentelli in EECS, David Aldous and Mike Jordan in Statistics. Saurabh Amin has proven to be a great collaborator on this research project, and part of the results presented here are his own. Alessandro D’Innocenzo, Aaron Ames, Minghua Chen and Ling Shi have shared many hours of discovery and exciting re- search with me. I would like to acknowledge the contribution of the control people in Cory 333, especially my fellows Luca Schenato, Bruno Sinopoli and John Koo. On a personal note, I would like to gratefully thank the people at Berkeley that I have had the privilege to spend my best time with, and which have greatly influenced my own growth during graduate school. In particular, a heartful thank you to the people of the IISA and, foremost, to Alvise Bonivento, Fabrizio Bisetti, Alberto Diminin, and Alessandro Pinto. The concluding acknowledgment, without adding any obvious and granted explanations or justifications for it, goes to my family and to Lynn. I know that they understand what I feel when I write these words: grazie.

    Alessandro Abate Berkeley, CA, Summer/Fall 2007

    i

  • Contents

    Acknowledgements i

    List of Figures iv

    Introduction 1

    1 Modeling 6

    1.1 Deterministic Hybrid System Model . . . . . . . . . . . . . . . . . . . 6

    1.2 Probabilistic Generalizations . . . . . . . . . . . . . . . . . . . . . . . 13

    1.2.1 General Stochastic Hybrid System Model . . . . . . . . . . . . 17

    1.2.2 Process Semigroup and Generator . . . . . . . . . . . . . . . . 23

    1.3 Elimination of Guards . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    1.3.1 Approximation Procedure . . . . . . . . . . . . . . . . . . . . 28

    1.3.2 Convergence Properties . . . . . . . . . . . . . . . . . . . . . . 30

    1.3.3 Convenience of the new form: a claim . . . . . . . . . . . . . 36

    1.4 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    1.4.1 Hybrid Dynamics through Random Measures . . . . . . . . . 45

    1.4.2 State-Dependent Thinning Procedure . . . . . . . . . . . . . . 54

    1.4.3 A Discretization Scheme with Convergence . . . . . . . . . . . 56

    2 Probabilistic Reachability and Safety 59

    2.1 The Concept of Reachability in Systems and Control Theory . . . . . 59

    2.1.1 Discrete-Time Controlled Stochastic Hybrid Systems . . . . . 64

    2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    2.2.1 Definition of the Concept . . . . . . . . . . . . . . . . . . . . . 70

    ii

  • 2.2.2 Probabilistic Reachability Computations . . . . . . . . . . . . 73

    2.2.3 Maximal Probabilistic Safe Sets Computation . . . . . . . . . 78

    2.2.4 Extensions to the Infinite Horizon Case . . . . . . . . . . . . . 86

    2.2.5 Regulation and Practical Stabilization Problems . . . . . . . . 90

    2.2.6 Other Related Control Problems . . . . . . . . . . . . . . . . . 91

    2.2.7 Embedding Performance into the Problem . . . . . . . . . . . 91

    2.2.8 State and Control Discretization . . . . . . . . . . . . . . . . . 92

    2.2.9 Connections with the Literature . . . . . . . . . . . . . . . . . 101

    2.3 Computations: A Benchmark Case Study . . . . . . . . . . . . . . . . 105

    2.3.1 Case Study: Temperature Regulation - Modeling . . . . . . . 105

    2.3.2 Case Study: Temperature Regulation - Control Synthesis . . . 108

    2.3.3 Numerical Approximations . . . . . . . . . . . . . . . . . . . . 124

    2.3.4 Mitigating the Curse of Dimensionality . . . . . . . . . . . . . 126

    2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

    2.4.1 Survival Analysis of Bacillus subtilis . . . . . . . . . . . . . . 133

    2.4.2 A Model for Antibiotic Synthesis

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