Robustness of stochastic discrete-time switched linear systems

Transcription

1 Robustness of stochastic discrete-time switched linear systems Hycon2-Balcon Workshop on Control Systems & Technologies for CPS Belgrad 2-3 July 2013 with application to control with shared resources L2S Supélec

2 Introduction Common resource shared among many users Embedded Systems Networked Systems Resource: Computation time Users: Concurrent (control) tasks Resource: Channel bandwidth Users: Communicating nodes 2

3 Introduction Common resource shared among many users Embedded Systems Networked Systems Resource: Computation time Users: Concurrent (control) tasks Resource: Channel bandwidth Users: Communicating nodes Time patterns for the access to the shared resource can widely vary fluctuations in computation time variability in transmission instants 2

4 Introduction Common resource shared among many users Embedded Systems Networked Systems Resource: Computation time Users: Concurrent (control) tasks Resource: Channel bandwidth Users: Communicating nodes Time patterns for the access to the shared resource can widely vary fluctuations in computation time variability in transmission instants Many classical (stability) results are based on Worst-Case approaches: deterministic bounds 2

5 Introduction Common resource shared among many users Embedded Systems Networked Systems Resource: Computation time Users: Concurrent (control) tasks Resource: Channel bandwidth Users: Communicating nodes Time patterns for the access to the shared resource can widely vary fluctuations in computation time variability in transmission instants Many classical (stability) results are based on Worst-Case approaches: deterministic bounds Stochastic models allow to better exploit the knowledge of the time patterns 2

6 Motivating examples - Embedded Systems Time allotted for computation At each step it varies depending of the previous step 3

7 Motivating examples - Embedded Systems Time allotted for computation p 11 s 1 s 2 p 13 p 12 p 21 p 32 p 31 s 3 p 33 p 23 p 22 s i : allotted time = t i At each step it varies depending of the previous step Markov Chain 3

8 Motivating examples - Embedded Systems Time allotted for computation p 11 s 1 s 2 p 13 p 12 p 21 p 32 p 31 s 3 p 33 p 23 p 22 s i : allotted time = t i At each step it varies depending of the previous step Markov Chain Time required for computation It can vary according to some distribution of times 3

9 Motivating examples - Embedded Systems Time allotted for computation p 11 s 1 s 2 p 13 p 12 p 21 p 32 p 31 s 3 p 33 p 23 p 22 s i : allotted time = t i At each step it varies depending of the previous step Markov Chain Time required for computation P j-th control task It can vary according to some distribution of times i.i.d. process t c 3

10 Motivating examples - Embedded Systems Time allotted for computation p 11 s 1 s 2 p 13 p 12 p 21 p 32 p 31 s 3 p 33 p 23 p 22 s i : allotted time = t i At each step it varies depending of the previous step Markov Chain Time required for computation P j-th control task It can vary according to some distribution of times i.i.d. process t c Which is the control task scheduled at each step? 3

11 Motivating examples - Networked Systems Available bandwidth At each step the network load condition varies depending of the previous step 4

12 Motivating examples - Networked Systems Available bandwidth p 11 s 1 s 2 p 13 p 12 p 21 p 32 p 31 s 3 p 33 p 23 p 22 s i : i-th occupation status At each step the network load condition varies depending of the previous step Markov Chain 4

13 Motivating examples - Networked Systems Available bandwidth p 11 s 1 s 2 p 13 p 12 p 21 p 32 p 31 s 3 p 33 p 23 p 22 s i : i-th occupation status At each step the network load condition varies depending of the previous step Markov Chain Transmission delay of a packet It can vary according to a different distribution for each network condition 4

14 Motivating examples - Networked Systems Available bandwidth p 11 s 1 s 2 p 13 p 12 p 21 p 32 p 31 s 3 p 33 p 23 p 22 s i : i-th occupation status At each step the network load condition varies depending of the previous step Markov Chain Transmission delay of a packet P j-th network condition It can vary according to a different distribution for each network condition i.i.d. process d t 4

15 Motivating examples - Networked Systems Available bandwidth p 11 s 1 s 2 p 13 p 12 p 21 p 32 p 31 s 3 p 33 p 23 p 22 s i : i-th occupation status At each step the network load condition varies depending of the previous step Markov Chain Transmission delay of a packet P j-th network condition It can vary according to a different distribution for each network condition i.i.d. process d t Which is the delay experienced by a packet sent at a step k? 4

16 Markov Jump Linear System model where Discrete-time finite-state homogeneous Markov chain (FSH-MC) 5

17 Markov Jump Linear System model where Discrete-time finite-state homogeneous Markov chain (FSH-MC) where 5

18 Classical stochastic stability definitions Given a constant δ-gas the MJLS is said to be: δ-moment global asymptotically stable if for any and any initial distribution δ-ges δ-moment global exponentially stable if there exist such that for any and any initial distribution as-gs almost surely globally stable if for any any initial distribution and 6

19 Stochastic Jump Linear System model (+ disturbance) where stochastic process 7

20 Stochastic Jump Linear System model (+ disturbance) where stochastic process where FSH-MC taking values in 7

21 Stochastic Jump Linear System model (+ disturbance) where stochastic process deterministic, locally bounded function where FSH-MC taking values in 7

22 Robust stochastic stability definitions Given and a constant δ-iss, the SJLS is said to be: δ-moment input-to-state stable if there exist s.t. for any, any δ-eiss δ-moment exponentially input-state stable if there exist and s.t. for any and any as-iss almost surely input-to-state stable if there exist s.t. for any, any 8

23 Equivalence and implications Theorem For the system driven by the stochastic process distributions are described by the evolution, whose δ-gas, δ-ges, δ-iss and δ-eiss are equivalent. Moreover, any of them implies as-iss. 9

24 Conditions for δ-stabilities and 2-stabilities Theorem The aforementioned SJLS is δ-eiss if there exist M matrices and such that for all and for all 10

25 Conditions for δ-stabilities and 2-stabilities Theorem The aforementioned SJLS is δ-eiss if there exist M matrices and such that for all and for all It is 2-EISS if there exist M matrices that such where 10

26 Conditions for as-iss Theorem The aforementioned SJLS is as-iss if there exist M matrices such that one of the following conditions is verified 11

27 Example TORA system and Anytime control (1) 12

28 Example TORA system and Anytime control (1) 1 ( z) 2 ( z) d i 2 i 2 r + y F 1 ( z) F 2 ( z) G(z) n i 1 i 1 Anytime control: the more you compute the better you control i K 1 ( z) K 2 ( z) 1 ( z) 2 ( z) 12

29 Example TORA system and Anytime control (1) 1 ( z) 2 ( z) d i 2 i 2 r + y F 1 ( z) F 2 ( z) G(z) n i 1 i 1 Anytime control: the more you compute the better you control i K 1 ( z) K 2 ( z) 1 ( z) 2 ( z) 12

30 Example TORA system and Anytime control (2) FSH Markov Chain 13

31 Example TORA system and Anytime control (2) FSH Markov Chain P P i.i.d. processes T 1 T 2 13

32 Example TORA system and Anytime control (3) 1 ( 1, 1 ) 0.8 [rad] Time [s] 1 ( 2, 2 ) 0.8 [rad] Time [s] 14

33 Example TORA system and Anytime control (3) 1 ( 1, 1 ) 0.8 [rad] SJLS Time [s] [rad] ( 2, 2 ) [rad] Time [s] 2-EISS system Time [s] 14

34 Conclusions and future works A model of SJLS encompassing switching systems driven by MCs and i.i.d. processes has been introduced; 15

35 Conclusions and future works A model of SJLS encompassing switching systems driven by MCs and i.i.d. processes has been introduced; Stochastic ISS definitions have been presented. They smoothly connect deterministic ISS results and classical stochastic stability definitions for SJLSs; 15

36 Conclusions and future works A model of SJLS encompassing switching systems driven by MCs and i.i.d. processes has been introduced; Stochastic ISS definitions have been presented. They smoothly connect deterministic ISS results and classical stochastic stability definitions for SJLSs; Easily testable sufficient conditions have been provided. 15

37 Conclusions and future works A model of SJLS encompassing switching systems driven by MCs and i.i.d. processes has been introduced; Stochastic ISS definitions have been presented. They smoothly connect deterministic ISS results and classical stochastic stability definitions for SJLSs; Easily testable sufficient conditions have been provided. In the future we aim at further extending the stochastic model to catch the complex behaviors arising when the interaction of different SJLSs on the shared resource induces a correlation among the stochastic properties of the SJLSs; 15

38 Conclusions and future works A model of SJLS encompassing switching systems driven by MCs and i.i.d. processes has been introduced; Stochastic ISS definitions have been presented. They smoothly connect deterministic ISS results and classical stochastic stability definitions for SJLSs; Easily testable sufficient conditions have been provided. In the future we aim at further extending the stochastic model to catch the complex behaviors arising when the interaction of different SJLSs on the shared resource induces a correlation among the stochastic properties of the SJLSs; Quantitative performance metrics will be investigated as well. 15

39 Robustness of stochastic discrete-time switched linear systems Hycon2-Balcon Workshop on Control Systems & Technologies for CPS Belgrad 2-3 July 2013 with application to control with shared resources L2S Supélec