Networked Control System Protocols Modeling & Analysis using Stochastic Impulsive Systems

Transcription

1 Networked Control System Protocols Modeling & Analysis using Stochastic Impulsive Systems João P. Hespanha Center for Control Dynamical Systems and Computation

2 Talk outline Examples feedback over shared communication network estimation using remote sensor Analysis tools Stochastic Hybrid Systems driven by renewal processes Lyapunov-based analysis of Stochastic Hybrid Systems (ex) students: D. Antunes (IST), Y. Xu (Advertising.com) collaborators: C. Silvestre (IST) acknowledgements: NSF, AFOSR (STTR program) disclaimer: This is an overview, technical details in papers referenced in bottom right corner

3 Deterministic Hybrid Systems continuous dynamics guard conditions reset-maps

4 Stochastic Hybrid Systems continuous dynamics transition intensities (probability of transition in interval (t, t+dt]) reset-maps

5 Stochastic Hybrid Systems continuous dynamics transition intensities (probability of transition in interval (t, t+dt]) reset-maps Special case: When all λ are constant x(t) is a Markov process & times between jumps are exponentially distributed q = 1 q = 2 q = 3 closely related to the so called Markovian Jump Systems [Costa, Fragoso, Boukas, Loparo, Lee, Dullerud]

6 Example I: Networked Control System controller process hold 1 hold 2 sampling times sensor 1 shared network sensor 2 process: controller: round-robin network access: hold

7 Example I: Networked Control System controller process hold 1 hold 2 sampling times sensor 1 shared network sensor 2 process: controller: round-robin What if the network access: is not available at a sample time t k? 1 st wait until network becomes available 2 nd send (old) data from original sampling of continuous-time output or hold 2 nd send (latest) data from current sampling of continuous-time output ) intersampling times t k+1 t k typically become random variables

8 Example I: Networked Control System controller process hold 1 hold 2 sampling times sensor 1 shared network sensor 2 Suppose t k+1 t k» i.i.d., exponentially distributed

9 Example I: Networked Control System controller process very unrealistic! sampling hold at best, 1 t k+1 t k» i.i.d., constant times + exponential sensor 1 shared but then x(t) is not a Markov process hold 2 network sensor 2 Suppose t k+1 t k» i.i.d., exponentially distributed

10 Example I: Networked Control System controller process hold 1 hold 2 sampling times sensor 1 shared network sensor 2 Suppose t k+1 t k» i.i.d., with cumulative distribution function F( ) τ(t) t 1 t 2 t 3 t the aggregate state (x,τ) is a Markov process

11 Impulsive Syst. driven by Renewal Proc. impulsive system same continuous dynamics for all modes N(t) # of jumps before time t renewal process (iid inter-increment times)

12 Impulsive Syst. driven by Renewal Proc. impulsive system same continuous dynamics for all modes N(t) # of jumps before time t renewal process (iid inter-increment times) Theorem: (for simplicity, conditions stated for equal reset matrices: J 1 =J 2 2 R n n ) system is stochastically stable, i.e., m LMI on P n n expected value w.r.t. inter-jump times and or Kronecker product spectral radius condition on n 2 n 2 matrix [ACC 09]

13 Talk outline Examples feedback over shared communication network estimation using remote sensor Analysis tools Stochastic Hybrid Systems driven by renewal processes Lyapunov-based analysis of Stochastic Hybrid Systems (ex) students: D. Antunes (IST), Y. Xu (Advertising.com) collaborators: C. Silvestre (IST) acknowledgements: NSF, AFOSR (STTR program) disclaimer: This is an overview, technical details in papers referenced in bottom right corner

14 Example II: Estimation through network process state-estimator x white noise disturbance encoder x(t 1 ) x(t 2 ) packet-switched network decoder for simplicity: full-state available no measurement noise no quantization no transmission delays encoder logic determines when to send measurements to the network decoder logic determines how to incorporate received measurements

15 Stochastic communication logic process state-estimator x encoder white noise disturbance x(t 1 ) x(t 2 ) packet-switched network encoder logic determines when to send measurements to the network decoder for simplicity: full-state available no measurement noise no quantization no transmission delays decoder logic determines how to incorporate received measurements [similar ideas pursued by Astrom, Tilbury, Hristu, Kumar, Basar]

16 process Example II: Remote estimation state-estimator x encoder Error dynamics: white noise disturbance x(t 1 ) x(t 2 ) packet-switched network decoder for simplicity: full-state available no measurement noise no quantization no transmission delays prob. of sending data in [t,t+dt) depends on current error e reset error to zero [CDC 04, CRC Press 06]

17 Analysis Lie derivative Given scalar-valued function à : R n [0,1)! R derivative along solution to ODE L f à Lie derivative of à Basis of Lyapunov formal arguments to establish boundedness and stability E.g., picking x(t) remains bounded along trajectories!

18 Generator of a SHS prob. of sending data in [t,t+dt) depends on current error e reset error to zero Given scalar-valued function à : Q R n [0,1)! R x & q are discontinuous, but the expected value is differentiable!!! where Dynkin s formula (in differential form) generator for the SHS instantaneous variation intensity Lie derivative reset term diffusion term Disclaimer: see Nonlinear Analysis 05 for technical assumptions

19 Generator of a SHS prob. of sending data in [t,t+dt) depends on current error e Given scalar-valued function à : Q R n [0,1)! R where reset error to zero generalizes to large classes of Stochastic Hybrid Systems x & q are discontinuous, but the expected value is differentiable!!! Dynkin s formula (in differential form) generator for the SHS instantaneous variation intensity Lie derivative reset term diffusion term Disclaimer: see Nonlinear Analysis 05 for technical assumptions

20 Lyapunov-based stability analysis error dynamics in remote estimation Dynkin s formula For constant rate: λ(e) = γ (exp. distributed inter-jump times) 1. E[ e ]! 0 if and only if γ > <[λ(a)] 2. E[ e m ] bounded if and only if γ > m <[λ(a)] getting more moments bounded requires higher comm. rates

21 Lyapunov-based stability analysis error dynamics in remote estimation Dynkin s formula For constant rate: λ(e) = γ (exp. distributed inter-jump times) 1. E[ e ]! 0 if and only if γ > <[λ(a)] 2. E[ e m ] bounded if and only if γ > m <[λ(a)] getting more moments bounded requires higher comm. rates For polynomial rates: λ(e) = (e 0 Q e) k Q > 0, k > 0 (reactive transmissions) 1. E[ e ]! 0 (always) 2. E[ e m ] bounded 8 m Moreover, one can achieve the same E[ e 2 ] with less communication than with a constant rate or periodic transmissions [CDC 04, Birkhauser 06]

22 Conclusions 1. A simple SHS model that finds use in several areas (networked control systems, network traffic modeling, biochemistry) 2. The analysis of SHSs is challenging but there are tools available (generator, Lyapunov methods, moment dynamics, truncations) 3. Lots of work to be done: theory stability/robustness/performance of SHS networked control systems protocol design to optimize performance & minimize communication resources