Formal Techniques for the Verification and Optimal Control of Probabilistic Systems in the Presence of Modeling Uncertainties
Alberto Puggelli

Citation
Alberto Puggelli. "Formal Techniques for the Verification and Optimal Control of Probabilistic Systems in the Presence of Modeling Uncertainties". PhD thesis, University of California, Berkeley, August, 2014.

Abstract
We present a framework to design and verify the behavior of stochastic systems whose parameters are not known with certainty but are instead affected by modeling uncertainties, due for example to modeling errors, non-modeled dynamics or inaccuracies in the probability estimation. Our framework can be applied to the analysis of intrinsically randomized systems (e.g., random back off schemes in wireless protocols) and of abstractions of deterministic systems whose dynamics are interpreted stochastically to simplify their representation (e.g., the forecast of wind availability). In the first part of the dissertation, we introduce the model of Convex Markov Decision Processes (Convex-MDPs) as the modeling framework to represent the behavior of stochastic systems. Convex-MDPs generalize MDPs by expressing state-transition probabilities not only with fixed realization frequencies but also with non-linear convex sets of probability distribution functions. These convex sets represent the uncertainty in the modeling process. In the second part of the dissertation, we address the problem of formally verifying properties of the execution behavior of Convex-MDPs. In particular, we aim to verify that the system behaves correctly under all valid operating conditions and under all possible resolutions of the uncertainty in the state-transition probabilities. We use Probabilistic Computation Tree Logic (PCTL) as the formal logic to express system properties. Using results on strong duality for convex programs, we present a model-checking algorithm for PCTL properties of Convex-MDPs, and prove that it runs in time polynomial in the size of the model under analysis. The developed algorithm is the first known polynomial-time algorithm for the verification of PCTL properties of Convex-MDPs. This result allows us to lower the previously known algorithmic complexity upper bound for Interval-MDPs from co-NP to P, and it is valid also for the more expressive (convex) uncertainty models supported by the Convex-MDP formalism. We apply the proposed framework and model-checking algorithm to the problem of formally verifying quantitative properties of models of the behavior of human drivers. We first propose a novel stochastic model of the driver behavior based on Convex Markov chains. The model is capable of capturing the intrinsic uncertainty in estimating the intricacies of the human behavior starting from experimentally collected data. We then formally verify properties of the model expressed in PCTL. Results show that our approach can correctly predict quantitative information about the driver behavior depending on his/her attention state, e.g., whether the driver is attentive or distracted while driving, and on the environmental conditions, e.g., the presence of an obstacle on the road. Finally, in the third part of the dissertation, we analyze the problem of synthesizing optimal control strategies for Convex-MDPs, aiming to optimize a given system performance, while guaranteeing that the system behavior fulfills a specification expressed in PCTL under all resolutions of the uncertainty in the state-transition probabilities. In particular, we focus on Markov strategies, i.e., strategies that depend only on the instantaneous execution state and not on the full execution history. We first prove that adding uncertainty in the representation of the state-transition probabilities does not increase the theoretical complexity of the synthesis problem, which remains in the class NP-complete as the analogous problem applied to MDPs, i.e., when all transition probabilities are known with certainty. We then interpret the strategy-synthesis problem as a constrained optimization problem and propose the first sound and complete algorithm to solve it. We apply the developed strategy-synthesis algorithm to the problem of generating optimal energy pricing and purchasing strategies for a for-profit energy aggregator whose portfolio of energy supplies includes renewable sources, e.g., wind. Economic incentives have been proposed to manage user demand and compensate for the intrinsic uncertainty in the prediction of the supply generation. Stochastic control techniques are however needed to maximize the economic profit for the energy aggregator while quantitatively guaranteeing quality-of-service for the users. We use Convex-MDPs to model the decision-making scenario and train the models with measured data, to quantitatively capture the uncertainty in the prediction of renewable energy generation. An experimental comparison shows that the control strategies synthesized using the proposed technique significantly increase system performance with respect to previous approaches presented in the literature.

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  • HTML 
    Alberto Puggelli. <a
href="http://www.terraswarm.org/pubs/354.html"
><i>Formal Techniques for the Verification and
Optimal Control of Probabilistic Systems in the Presence of
Modeling Uncertainties</i></a>, PhD thesis, 
University of California, Berkeley, August, 2014.
  • Plain text 
    Alberto Puggelli. "Formal Techniques for the
Verification and Optimal Control of Probabilistic Systems in
the Presence of Modeling Uncertainties". PhD thesis, 
University of California, Berkeley, August, 2014.
  • BibTeX 
    @phdthesis{Puggelli14_FormalTechniquesForVerificationOptimalControlOfProbabilistic,
    author = {Alberto Puggelli},
    title = {Formal Techniques for the Verification and Optimal
              Control of Probabilistic Systems in the Presence
              of Modeling Uncertainties},
    school = {University of California, Berkeley},
    month = {August},
    year = {2014},
    abstract = { We present a framework to design and verify the
              behavior of stochastic systems whose parameters
              are not known with certainty but are instead
              affected by modeling uncertainties, due for
              example to modeling errors, non-modeled dynamics
              or inaccuracies in the probability estimation. Our
              framework can be applied to the analysis of
              intrinsically randomized systems (e.g., random
              back off schemes in wireless protocols) and of
              abstractions of deterministic systems whose
              dynamics are interpreted stochastically to
              simplify their representation (e.g., the forecast
              of wind availability). In the first part of the
              dissertation, we introduce the model of Convex
              Markov Decision Processes (Convex-MDPs) as the
              modeling framework to represent the behavior of
              stochastic systems. Convex-MDPs generalize MDPs by
              expressing state-transition probabilities not only
              with fixed realization frequencies but also with
              non-linear convex sets of probability distribution
              functions. These convex sets represent the
              uncertainty in the modeling process. In the second
              part of the dissertation, we address the problem
              of formally verifying properties of the execution
              behavior of Convex-MDPs. In particular, we aim to
              verify that the system behaves correctly under all
              valid operating conditions and under all possible
              resolutions of the uncertainty in the
              state-transition probabilities. We use
              Probabilistic Computation Tree Logic (PCTL) as the
              formal logic to express system properties. Using
              results on strong duality for convex programs, we
              present a model-checking algorithm for PCTL
              properties of Convex-MDPs, and prove that it runs
              in time polynomial in the size of the model under
              analysis. The developed algorithm is the first
              known polynomial-time algorithm for the
              verification of PCTL properties of Convex-MDPs.
              This result allows us to lower the previously
              known algorithmic complexity upper bound for
              Interval-MDPs from co-NP to P, and it is valid
              also for the more expressive (convex) uncertainty
              models supported by the Convex-MDP formalism. We
              apply the proposed framework and model-checking
              algorithm to the problem of formally verifying
              quantitative properties of models of the behavior
              of human drivers. We first propose a novel
              stochastic model of the driver behavior based on
              Convex Markov chains. The model is capable of
              capturing the intrinsic uncertainty in estimating
              the intricacies of the human behavior starting
              from experimentally collected data. We then
              formally verify properties of the model expressed
              in PCTL. Results show that our approach can
              correctly predict quantitative information about
              the driver behavior depending on his/her attention
              state, e.g., whether the driver is attentive or
              distracted while driving, and on the environmental
              conditions, e.g., the presence of an obstacle on
              the road. Finally, in the third part of the
              dissertation, we analyze the problem of
              synthesizing optimal control strategies for
              Convex-MDPs, aiming to optimize a given system
              performance, while guaranteeing that the system
              behavior fulfills a specification expressed in
              PCTL under all resolutions of the uncertainty in
              the state-transition probabilities. In particular,
              we focus on Markov strategies, i.e., strategies
              that depend only on the instantaneous execution
              state and not on the full execution history. We
              first prove that adding uncertainty in the
              representation of the state-transition
              probabilities does not increase the theoretical
              complexity of the synthesis problem, which remains
              in the class NP-complete as the analogous problem
              applied to MDPs, i.e., when all transition
              probabilities are known with certainty. We then
              interpret the strategy-synthesis problem as a
              constrained optimization problem and propose the
              first sound and complete algorithm to solve it. We
              apply the developed strategy-synthesis algorithm
              to the problem of generating optimal energy
              pricing and purchasing strategies for a for-profit
              energy aggregator whose portfolio of energy
              supplies includes renewable sources, e.g., wind.
              Economic incentives have been proposed to manage
              user demand and compensate for the intrinsic
              uncertainty in the prediction of the supply
              generation. Stochastic control techniques are
              however needed to maximize the economic profit for
              the energy aggregator while quantitatively
              guaranteeing quality-of-service for the users. We
              use Convex-MDPs to model the decision-making
              scenario and train the models with measured data,
              to quantitatively capture the uncertainty in the
              prediction of renewable energy generation. An
              experimental comparison shows that the control
              strategies synthesized using the proposed
              technique significantly increase system
              performance with respect to previous approaches
              presented in the literature. },
    URL = {http://terraswarm.org/pubs/354.html}
}

Posted by Alberto Puggelli on 1 Sep 2014.
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