Scheduling of Event-Triggered Controllers on a Shared Network Anton Cervin and Toivo Henningsson

Transcription

1 on a Shared Network Automatic Control LTH Lund University Sweden

2 The Problem Plant 1 Plant 2 A 1 S 1 A 2 S 2 S 3 A 3 Plant 3 How to schedule the sampling and communication to get optimum control performance?

3 Keep It Simple First-order plants Impulse control Controller co-located with either sensor or actuator Shared network model: Idle Sampling event TimeT elapsed Busy

4 Comparison Time-triggered sampling and communication Classical sampled-data control TDMA (Time Division Multiple Access) Event-triggered sampling and communication Sporadic event-based control FDMA (Frequency Division Multiple Access) or CSMA (Carrier Sense Multiple Access)

5 Control Problem N first-order stochastic systems: dx i (t)=a i x i (t)dt+u i (t)dt+ σ i dw i (t), i=1,...,n We want to minimize where J= N i=1 J i 1 t J i =lim t t E xi 2 (s)ds 0

6 Control Problem Samples taken and sent at time instantst i [k]: x i [k]=x i (t i [k]) Impulse controls applied after fixed communication delayt: u i (t)= ( ) δ t t i [k] T u i [k] k Optimal control policy: u i [k]= e a it x i [k]

7 Time-Triggered Sampling 4 2 x Time Plantisampled at pre-determined (but possibly irregular) time instantst i [k] Gaussian state distribution at each point in time Trivial to evaluate cost

8 Event-Triggered Sampling 4 2 x Time Threshold-triggered sampling [Åström & Bernhardsson, 1999] Plantisampled if x i (t) r i and at leastt seconds since last sample Sporadic constraint models busy medium [Johannesson, Henningsson, Cervin, 2007] Non-Gaussian state distribution

9 Optimal Co-Design under TDMA t Static cyclic schedule with fixed slot sizet Time-triggered sampling Decision variables: Schedule lengthl Sequence{s 1,...,s l } [Rehnbinder & Sanfridson, 2001]

10 Optimal Co-Design under FDMA Each loop receives a fixed sharew i of the bandwidth Event-triggered sampling Control delay for loopi:t/w i Decision variables: Bandwidth shares{w i } Trigger thresholds{r i } t

11 (Sub-)Optimal Co-Design under CSMA Event-triggered sampling Sampling only if network free and access gained Decision variables: Trigger thresholds{r i } Priority function for the medium access Random access (Ethernet, WLAN) Static priorities (CAN) Dynamic priorities (Maximum Error First [Walsh et al., 1999]) t

12 Evaluation of Cost under CSMA Discrete-time approx.: fast sensor sampling witht s T State evolution between fast sensor sampling instants: e a it s x i (t k )+w i,k (T s ), x i (t k ) <r i x i (t k+1 )= w i,k (T), x i (t k ) r i & won e ait x(t k )+w i,k (T), x i (t k ) r i & lost Plant state distributions not independent. Options: Computation of N-dimensional state probability function Monte Carlo simulations

13 Performance Evaluation 1 N identical integrator plants with σ i =1 T = 1/N (total network bandwidth scales N) FDMA: w i =1/N TDMA: s={1,...,n} CSMA-rand Random access CSMA-dynprio Maximum error first

14 Optimal Cost Per Plant TDMA FDMA CSMA rand CSMA dynprio 1.4 Cost per plant Number of plants N

15 Optimal Thresholds under CSMA CSMA rand CSMA dynprio 1 Optimal threshold Number of plants N

16 Optimal Global Scheduling Policy? Control of two integrators: 3 Best local policy (CSMA dynprio) Optimal global policy 2 Control 2 x2 1 Control 1 Idle x 1 <1% difference in performance Optimal global policy not implementable

17 Performance Evaluation 2 Three different plants to be controlled: a 1,2,3 ={ 0.5,0,0.5}, T=1/3, σ i =1 Results after extensive optimization: Policy Optimal Resource Allocation Total Cost TDMA s={3,2,3,2,3,1} 2.56 FDMA w 1,2,3 ={0,0.397,0.603} 3.49 CSMA-rand r 1,2,3 ={1.12,0.92,0.77} 1.96 CSMA-statprio r 1,2,3 ={0.95,0.87,0.77} 1.95

18 Conclusions Event-triggered sampling and communication is more efficient The advantage increases as the number of loops increases WhenN, we approach the performance of Åström & Bernhardsson s aperiodic controller The medium access priority function is not very important Optimal design requires extensive numerical computations even for first-order systems Smarter approximations / better numerical methods needed

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