Stability analysis of linear systems under asynchronous samplings

Transcription

1 Haifa, April, 4 th, 2010 Stability analysis of linear systems under asynchronous samplings Alexandre Seuret CNRS Associate Researcher NeCS Team GIPSA-Lab / INRIA Rhônes-Alpes Grenoble, France 1

2 Motivation : Stability of sampled-data systems A simple example with a sampled input Hybrid problem : both continuous and discrete-time problem Continuous Process A/D Converter D/A + 0-holder Converter Controller In a realistic case, Problems to solve: To ensure the stability of the process; Provide less convervative conditions to obtain accurate bounds for T 2

3 Motivation : Stability of sampled-data systems Application to flight control implementation PhD : P. Andrianiaina Design of the sampling period? Probability x(tk) u(tk-tex) Reduced sampling period MCET Minimum Case Exec. Time Tex Execution Time WCET Worst Case Execution Time Research of less conservative implementation Guaranty of stability wrt. dead-line miss End-to-end control quality Security of the controller Robustness of the closed-loop system

4 Motivation : Stability of sampled-data systems Application to flight control implementation Hyp. : Tslot=WCET T T Sampling of sensors data sk x(sk) sk+1 x(sk+1) t Computation time dedicated to the control WCET WCET t Effective computation time Latency t Implementation of the control law in the actuators tk uimpl(tk)=u(x(sk)) tk+1 uimpl(tk+1)=u(x(sk)) t Main Objective: To reduce the lantency

5 Motivation : Stability of sampled-data systems Application to flight control implementation Hyp. : Tslot<WCET T T Sampling of sensors data sk x(sk) sk+1 x(sk+1) t Computation time dedicated to the control Tslot Tslot t Computation miss Effective computation time t Implementation of the control law in the actuators tk uimpl(tk)=u(x(sk)) tk+1 uimpl(tk+1)=u(x(sk)) t Main Objectives: Define the optimal Tslot Robustness wrt. input misses

6 Motivation : Stability of sampled-data systems Application to flight control implementation Following the previous scheme, the closed loop system is given by: is the effective control law in the actuators Constant sampling period: Constant delay: Sampled and delayed systems

7 Motivation : Stability of sampled-data systems Control a system through a network inducing delays Control objectives: Robustness wrt. Delays and samplings asynchrony x x x and Packet dropouts Schedule of the sampling period 7

8 Stability analysis of linear systems under asynchronous samplings Outline: Discrete-time approaches for sampled-data systems Continuous-time approaches for sampled-data systems Introduction of a mixed analysis Conclusions and perspectives 8

9 Stability analysis of linear systems under asynchronous samplings Outline: Discrete-time approach to sampled-data systems Continuous-time approach to sampled-data systems Introduction of a mixed analysis Conclusions and perspectives 9

10 1) Discrete-time approach A) Stability for systems with periodic sampling Consider: Discrete-time model: Solving the differential equation defining the system, we obtain where Then define 10

11 1) Discrete-time approach Asymptotic stability of sampled-data systems If is constant : Asymp. Stability conditions : Stability conditions based on Lyapunov theory The discrete-time system is stable if there exists P>0 such that: (1) Proof: Consider a Lyapunov function (1) ensures that 11

12 1) Discrete-time approach Asymptotic stability of sampled-data systems If is constant : Asymptotic stability 12

13 1) Discrete-time approach Asymptotic stability of sampled-data systems If Tk is time-varying Consider T=1 T=1.8 T=3.5 T=2.1 T=1, 3.5 T=1.8,

14 1) Discrete-time approach Asymptotic stability of sampled-data systems If Tk is time-varying, the Lyapunov approach still works Stability conditions based on Lyapunov theory The discrete-time system is stable if there exists P>0 such that: (1) Remark: Works well for a limited number of sampling periods. Limitations: The matrice P should be the same for all sampling periods. This leads to a large conservatism 14

15 1) Discrete-time approach Asymptotic stability of sampled-data systems If Tk is time-varying Some ideas to solve this problem Polytopic representations Discretization method 15

16 1) Discrete-time approach Asymptotic stability of sampled-data systems If Tk is time-varying Some ideas to solve this problem: Switched systems approach Sketch of the proof: (Hetel 2006) Define a finite set of Lyapunov functions such that for (i,j) in {1,,N} to avoid 16

17 1) Discrete-time approach Limitations Consider systems of the form: Time-varying systems Parameter uncertainties Input saturations Stability conditions are difficult to obtain since, in F, A and B appear in a non linear way. In such situations, continuous-time approaches appear relevant 17

18 Stability analysis of linear systems under asynchronous samplings Outline: Discrete-time approach to sampled-data systems Continuous-time approach to sampled-data systems Introduction of a mixed analysis Conclusions and perspectives 18

19 2) Continuous-time approach An input delay approach to sampled data systems Link with TDS? 19

20 2. Continuous-time approach An input delay approach to sampled data systems Main idea : Sampling = Time-varying delay [Mikheev et al. 88, Sobolev et al. 89. Aström et al. 92, Fridman et al 2004] Sampled-data signal (here with uniform sampling) Delayed signal with 20

21 2. Continuous-time approach An input delay approach to sampled data systems Objectives : Find Lyapunov-Krasovkii functionals 21

22 2. Continuous-time approach An input delay approach to sampled data systems [Fridman et al, Auto. 2004] Motivations: 1.Holds for both periodic and aperiodic samplings and also with additional input delay 2.Linear wrt. the system parameters A and B (TV systems, param. unc. and sat.) Disavantages: Conservative 22

23 Conservatism 2. Continuous-time approach Input delay approach: Lyapunov-Krasovskii functionals [Fridman et al 2004] [Nagahshtabrizi, SCL. 2008] [Seuret, CDC 2009, Fridman 2010 & Seuret TDS 2010] 23

24 3. Hybrid analysis Motivation: Difference between TDS and SDS Is there a link between a Lyapunov function for discrete-time systems and the Lyapunov-Krasovskii functional for continuous-time systems? More precisecly?? 24

25 1) Discrete-time approach Motivation: Difference between TDS and SDS Consider t =0.15 t=1 t=1.7 t=2.2 25

26 Stability analysis of linear systems under asynchronous samplings Outline: Discrete-time approach to sampled-data systems Continuous-time approach to sampled-data systems Introduction of a mixed analysis Conclusions and perspectives 26

27 3. Hybrid analysis [Seuret, provisionally accepted to automatica,2011] At the sampling instant: Interests is required to be positive definite No conditions on the positivity on Leads to stability conditions linear wrt. A,BK,T 27

28 3. Hybrid analysis Proof: Step 1 (ii)=>(i) 28

29 3. Hybrid analysis Proof: Step 2 (i)=>(ii) Proof: Step 3 : Inter-sampling behavior 29

30 3. Hybrid analysis Construction of such function and functionals No conditions on Submitted extensions: SOS tools [Peet,2011] Discretization [Briat,2011] Saturations [Gomès, 2011] 30

31 3. Hybrid analysis

32 3. Hybrid analysis Extension to exponential stability [Seuret, submitted to Automatica] Similar proof: Lyapunov functions (no delay) Exp. Lyapunov functionals Interests x a can be negative 2 xe t0 t t0 32 t

33 3. Hybrid analysis Extension to exponential stability Consider the same example: classical input delay approach the previous hybrid approach Stability Unstability 33

34 3. Hybrid analysis Extension to multi-rate sampling and packet loss Assumptions: Application to packet loss: 34

35 3. Hybrid analysis Extension to multi-rate sampling and packet loss [Seuret, submitted to Automatica] Sketch of the proof: Under the assumptions on the realisation of the probabilities, we obtain 35

36 3. Hybrid analysis Extension to multi-rate sampling and packet loss Another example: Multirate sampling: Packet losses: Unstable 36

37 Conclusions Summary: Review of different methods for the stability analysis of sampled-data systems Introduction of a novel methodology to the stability analysis of sampled-data systems. Future works Extensions to delayed systems 37