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Hespanha - Research Supported by NSF ARO Modeling and...

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Modeling and Analysis using Stochastic Hybrid System João Hespanha Research Supported by NSF & ARO
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Talk Outline (ex) students: D. Antunes (IST), A. Mesquita (UCSB), Y. Xu (Advertising.com), A. Singh (UCSD) collaborators: M. Khammash (UCSB), C. Silvestre (IST) acknowledgements: NSF, Institute for Collaborative bio-technologies (ARO), AFOSR (STTR program) disclaimer: This is an overview, technical details in papers referenced in bottom right corner… http://www.ece.ucsb.edu/~hespanha Modeling/Analysis tools Lyapunov-based analysis Moments dynamics Examples Biology / degradation regulation Biology / transcription regulation
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Talk Outline NSF, Institute for Collaborative bio-technologies (ARO), disclaimer: This is an overview, technical details in papers referenced in bottom right corner… http://www.ece.ucsb.edu/~hespanha Disclaimer: Several other important applications/researchers not mentioned in this talk. E.g., air traffic control [Bujorianu, Lygeros, Prandini, Hu, Tomlin,...] network traffic modeling [Bohacek, Lee, Yin, ...] queuing systems [Cassandras,...] economics [Davis, Yin,…] biology [Hu, Julius, Lygeros, Pappas,...] Examples Biology / degradation regulation Biology / transcription regulation Modeling/Analysis tools Lyapunov-based analysis Moments dynamics
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Deterministic Hybrid Systems guard conditions reset-maps continuous dynamics q ( t ) ! Q ={1,2,…} " discrete state x ( t ) ! R n " continuous state q 1 x f 1 x q 2 x f 2 x q 3 x f 3 x
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Deterministic Hybrid Systems guard conditions continuous dynamics q ( t ) ! Q ={1,2,…} " discrete state x ( t ) ! R n " continuous state ˙ x = f ( x ) g ( x ) 0? x φ ( x ) impulsive system (single discrete mode) q 1 x f 1 x q 2 x f 2 x q 3 x f 3 x reset-maps
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Stochastic Hybrid Systems transition intensities (probability of transition in small interval ( t , t + dt ] ) q ( t ) ! Q ={1,2,…} " discrete state x ( t ) ! R n " continuous state λ ( x ) dt probability of transition in an “elementary” interval ( t , t + dt ] instantaneous rate of transitions per unit of time λ ( x ) ! continuous dynamics Special case: When all " ! are constant time triggered SHS with exponential t k +1 " t k Markovian jump system q 1 x f 1 x q 2 x f 2 x q 3 x f 3 x reset-maps
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1. Initialize state: 2. Draw a unit-mean exponential random variable 3. Solve ODE until time t k+1 for which 4. Apply the corresponding reset map set k = k + 1 and go to 2. Construction of the Stochastic Process ˙ x = f ( x ) ˙ x = f ( x ) x ( t k ) = x k t t k λ ( x ) dt x φ ( x ) E exp(1) t k +1 t k λ ( x ( t )) dt E x ( t k +1 ) = x k +1 := φ ( x ( t k +1 )) x ( t 0 ) = x 0 k = 0 here we take x 0 as a given parameter
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Example #1: Degradation Regulation Degradation regulation " feedback mechanism used to regulate the concentration of a protein by destroying protein molecules “in excess” x ( t ) = concentration of protein X at time t production of X at constant rate x k dx degradation of “each protein X” at a constant rate x exp k d
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Example #1: Degradation Regulation Degradation regulation " feedback mechanism used to regulate the concentration of a protein by destroying protein molecules “in excess” x ( t ) = concentration of protein X at time t production of X at
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