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

### Alberto Puggelli

### EECS Department, University of California, Berkeley

### Technical Report No. UCB/EECS-2014-155

### August 15, 2014

### http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-155.pdf

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. 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. 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. 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, and on the environmental conditions. 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. 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 synthesized control strategies significantly increase system performance with respect to previous approaches presented in the literature.

**Advisors:** Alberto L. Sangiovanni-Vincentelli and Elad Alon

BibTeX citation:

@phdthesis{Puggelli:EECS-2014-155, Author= {Puggelli, Alberto}, Title= {Formal Techniques for the Verification and Optimal Control of Probabilistic Systems in the Presence of Modeling Uncertainties}, School= {EECS Department, University of California, Berkeley}, Year= {2014}, Month= {Aug}, Url= {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-155.html}, Number= {UCB/EECS-2014-155}, 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. 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. 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. 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, and on the environmental conditions. 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. 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 synthesized control strategies significantly increase system performance with respect to previous approaches presented in the literature.}, }

EndNote citation:

%0 Thesis %A Puggelli, Alberto %T Formal Techniques for the Verification and Optimal Control of Probabilistic Systems in the Presence of Modeling Uncertainties %I EECS Department, University of California, Berkeley %D 2014 %8 August 15 %@ UCB/EECS-2014-155 %U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-155.html %F Puggelli:EECS-2014-155