Event-triggered stabilization of linear systems under channel blackouts

Transcription

1 Event-triggered stabilization of linear systems under channel blackouts Pavankumar Tallapragada, Massimo Franceschetti & Jorge Cortés Allerton Conference, 30 Sept Acknowledgements: National Science Foundation (Grants CNS , CNS ) 1 / 16

2 Networked control systems Shared communication resource Time-varying communication rates Channel may not be available during some intervals (blackouts) Time-triggered strategies would be very conservative Event-triggered controllers typically assume on-demand availability of channel 1 1 An important exception: Anta, Tabuada (2009) 2 / 16

3 Networked control systems Shared communication resource Time-varying communication rates Channel may not be available during some intervals (blackouts) Time-triggered strategies would be very conservative Event-triggered controllers typically assume on-demand availability of channel 1 Quantization 1 An important exception: Anta, Tabuada (2009) 2 / 16

4 Networked control systems Shared communication resource Time-varying communication rates Channel may not be available during some intervals (blackouts) Time-triggered strategies would be very conservative Event-triggered controllers typically assume on-demand availability of channel 1 Quantization Key to online state based transmission policy: data capacity 1 An important exception: Anta, Tabuada (2009) 2 / 16

5 System description Plant dynamics: ẋ(t) = Ax(t) + Bu(t), u(t) = K ˆx(t), x(t) R n 3 / 16

6 System description Plant dynamics: ẋ(t) = Ax(t) + Bu(t), u(t) = K ˆx(t), x(t) R n Communication model: k (t k, p k ) b k R a(t k ) = p k R(t k ) # of bits transmitted at t k is b k = np k Can choose {t k }, {p k }, { r k } 3 / 16

7 System description Plant dynamics: ẋ(t) = Ax(t) + Bu(t), u(t) = K ˆx(t), x(t) R n Communication model: k (t k, p k ) b k R a(t k ) = p k R(t k ) # of bits transmitted at t k is b k = np k Can choose {t k }, {p k }, { r k } Dynamic controller flow: ˆx(t) = Aˆx(t) + Bu(t) = Āˆx(t), t [ r k, r k+1 ) 3 / 16

8 System description Plant dynamics: ẋ(t) = Ax(t) + Bu(t), u(t) = K ˆx(t), x(t) R n Communication model: k (t k, p k ) b k R a(t k ) = p k R(t k ) # of bits transmitted at t k is b k = np k Can choose {t k }, {p k }, { r k } Dynamic controller flow: ˆx(t) = Aˆx(t) + Bu(t) = Āˆx(t), t [ r k, r k+1 ) Dynamic controller jump: ˆx( r k ) q k (x(t k ), ˆx(t k )) Encoding error: x e x ˆx 3 / 16

9 Quantization Can design 2 consistent algorithms for the encoder and decoder to implement quantizer q k so that: 2 Tallapragada, Cortés (2016) 4 / 16

10 Quantization Can design 2 consistent algorithms for the encoder and decoder to implement quantizer q k so that: If the decoder knows d e (t 0 ) s.t. x e (t 0 ) d e (t 0 ) 2 Tallapragada, Cortés (2016) 4 / 16

11 Quantization Can design 2 consistent algorithms for the encoder and decoder to implement quantizer q k so that: If the decoder knows d e (t 0 ) s.t. x e (t 0 ) d e (t 0 ) Both encoder and decoder compute recursively: d e (t) e A(t t k) δ k, t [ r k, r k+1 ), k Z 0 δ k+1 = 1 2 p k+1 d e(t k+1 ). 2 Tallapragada, Cortés (2016) 4 / 16

12 Quantization Can design 2 consistent algorithms for the encoder and decoder to implement quantizer q k so that: If the decoder knows d e (t 0 ) s.t. x e (t 0 ) d e (t 0 ) Both encoder and decoder compute recursively: d e (t) e A(t t k) δ k, t [ r k, r k+1 ), k Z 0 δ k+1 = 1 2 p k+1 d e(t k+1 ). Then, x e (t) d e (t), for all t t 0 2 Tallapragada, Cortés (2016) 4 / 16

13 Objective Suppose Ā = A + BK is Hurwitz P Ā + ĀT P = Q Lyapunov function: x V (x) = x T P x 5 / 16

14 Objective Suppose Ā = A + BK is Hurwitz P Ā + ĀT P = Q Lyapunov function: x V (x) = x T P x Desired performance function: V d (t) = V d (t 0 )e β(t t 0) Performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 5 / 16

15 Objective Suppose Ā = A + BK is Hurwitz P Ā + ĀT P = Q Lyapunov function: x V (x) = x T P x Desired performance function: V d (t) = V d (t 0 )e β(t t 0) Performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 Design objective: Design event-triggered communication policy that is applicable to channels with time-varying rates and blackouts Recursively determine {t k }, {p k } and { r k } Ensure a uniform positive lower bound for {t k t k 1 } k Z>0 5 / 16

16 Time-slotted channel model p t R t R(t) = R j, t (θ j, θ j+1 ], min comm. rate: p k (t k, p k ) R(t k) p(t) = π j, t (θ j, θ j+1 ], max packet size: p k p(t k ) j th time-slot is of length T j = θ j+1 θ j Channel is not available when p = 0 (channel blackout) Channel evolution is known a priori 6 / 16

17 Time-slotted channel model p t R t R(t) = R j, t (θ j, θ j+1 ], min comm. rate: p k (t k, p k ) R(t k) p(t) = π j, t (θ j, θ j+1 ], max packet size: p k p(t k ) j th time-slot is of length T j = θ j+1 θ j Channel is not available when p = 0 (channel blackout) Channel evolution is known a priori Main idea of solution: make sure the encoding error is sufficiently small at the beginning of a channel blackout 6 / 16

18 Time-slotted channel model p t R t R(t) = R j, t (θ j, θ j+1 ], min comm. rate: p k (t k, p k ) R(t k) p(t) = π j, t (θ j, θ j+1 ], max packet size: p k p(t k ) j th time-slot is of length T j = θ j+1 θ j Channel is not available when p = 0 (channel blackout) Channel evolution is known a priori Main idea of solution: make sure the encoding error is sufficiently small at the beginning of a channel blackout Need to quantify data capacity 6 / 16

19 Data capacity max # of bits that can be communicated during the time interval [τ 1, τ 2 ], overall all possible {t k } and {p k } D(τ 1, τ 2 ) max {t k },{p k } s.t.... n k τ2 k=k τ1 p k k τ1 = 3, k τ2 = 7 7 / 16

20 Data capacity max # of bits that can be communicated during the time interval [τ 1, τ 2 ], overall all possible {t k } and {p k } D(τ 1, τ 2 ) max {t k },{p k } s.t.... n k τ2 k=k τ1 p k k τ1 = 3, k τ2 = 7 Equivalent to optimal allocation of discrete # bits to be transmitted in each time slot 7 / 16

21 Data capacity as allocation problem Max #{ bits that may be transmitted in slot j nr j T j + n π j, if π j > 0 nφ j 0, if π j = 0 p t 8 / 16

22 Data capacity as allocation problem Max #{ bits that may be transmitted in slot j nr j T j + n π j, if π j > 0 nφ j 0, if π j = 0 Available { time in slot j is affected by prior transmissions nr j Tj (φ j f j nφ j 0 ) + n π j, if T j (φ j f j 0 ) > 0 0 otherwise p t 8 / 16

23 Data capacity as allocation problem Max #{ bits that may be transmitted in slot j nr j T j + n π j, if π j > 0 nφ j 0, if π j = 0 Available { time in slot j is affected by prior transmissions nr j Tj (φ j f j nφ j 0 ) + n π j, if T j (φ j f j 0 ) > 0 0 otherwise Count { only the bits also received φ j Tj (φ j f j R j 0 ) + θ jf θ j+1, if T j (φ j f j 0 ) > 0 0, otherwise. p t 8 / 16

24 Data capacity as allocation problem Max #{ bits that may be transmitted in slot j nr j T j + n π j, if π j > 0 nφ j 0, if π j = 0 Available { time in slot j is affected by prior transmissions nr j Tj (φ j f j nφ j 0 ) + n π j, if T j (φ j f j 0 ) > 0 0 otherwise Count { only the bits also received φ j Tj (φ j f j R j 0 ) + θ jf θ j+1, if T j (φ j f j 0 ) > 0 0, otherwise. D(θ j0, θ jf ) = max φ j Z 0 s.t.... n j f 1 j=j 0 φ j. p t 8 / 16

25 A suboptimal solution for slowly varying channels Proposition Assume π j < T j+1, j N j f j R 0 j received before the end of slot j + 1). (any bits transmitted in slot j are 9 / 16

26 A suboptimal solution for slowly varying channels Proposition Assume π j < T j+1, j N j f j R 0 j (any bits transmitted in slot j are received before the end of slot j + 1). Let φ r = argmax φ j R 0 Let s.t.... j f 1 j=j 0 φ j (LP). φ N φ r ( φ r j 0,..., φ r j f 1 ), D s (θ j0, θ jf ) n j f 1 j=j 0 φ N j. 9 / 16

27 A suboptimal solution for slowly varying channels Proposition Assume π j < T j+1, j N j f j R 0 j (any bits transmitted in slot j are received before the end of slot j + 1). Let φ r = argmax φ j R 0 Let s.t.... j f 1 j=j 0 φ j (LP). φ N φ r ( φ r j 0,..., φ r j f 1 ), D s (θ j0, θ jf ) n j f 1 j=j 0 φ N j. Then φ N is a sub-optimal solution D(θ j0, θ jf ) D s (θ j0, θ jf ) n(j f 1 j 0 ). 9 / 16

28 Real time computation of data capacity Proposition Let φ (or φ N ) be any optimizing solution to D(θ j0, θ jf ) (or D s (θ j0, θ jf )). 10 / 16

29 Real time computation of data capacity Proposition Let φ (or φ N ) be any optimizing solution to D(θ j0, θ jf ) (or D s (θ j0, θ jf )). For any t [θ j0, θ j0 +1) (any t in j 0 slot) ˆD(t, θ jf ) [ n φ j 0 R j0 (t θ j0 ) ] + + n j f 1 j=j 0 +1 ˆD s (t, θ jf ) [ n φ N j 0 R j0 (t θ j0 ) ] + + n j f 1 j=j 0 +1 φ j φ N j, 10 / 16

30 Real time computation of data capacity Proposition Let φ (or φ N ) be any optimizing solution to D(θ j0, θ jf ) (or D s (θ j0, θ jf )). For any t [θ j0, θ j0 +1) (any t in j 0 slot) ˆD(t, θ jf ) [ n φ j 0 R j0 (t θ j0 ) ] + + n j f 1 j=j 0 +1 ˆD s (t, θ jf ) [ n φ N j 0 R j0 (t θ j0 ) ] + + n j f 1 j=j 0 +1 φ j φ N j, Then, 0 D(t, θ jf ) ˆD(t, θ jf ) n and 0 D s (t, θ jf ) ˆD s (t, θ jf ) n. 10 / 16

31 Real time computation of data capacity Proposition Let φ (or φ N ) be any optimizing solution to D(θ j0, θ jf ) (or D s (θ j0, θ jf )). For any t [θ j0, θ j0 +1) (any t in j 0 slot) ˆD(t, θ jf ) [ n φ j 0 R j0 (t θ j0 ) ] + + n j f 1 j=j 0 +1 ˆD s (t, θ jf ) [ n φ N j 0 R j0 (t θ j0 ) ] + + n j f 1 j=j 0 +1 φ j φ N j, Then, 0 D(t, θ jf ) ˆD(t, θ jf ) n and 0 D s (t, θ jf ) ˆD s (t, θ jf ) n. Significance: Sufficient to solve the data capacity problem for intervals [θ j0, θ jf ] of interest. 10 / 16

32 Elements of the event-trigger Recall performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 11 / 16

33 Elements of the event-trigger Recall performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 Channel trigger function: h ch (t) ɛ(t) de(t) ρ T (h pf (t)), ɛ(t) c V d (t) 11 / 16

34 Elements of the event-trigger Recall performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 Channel trigger function: h ch (t) Lemma If h pf (t) 1 and h ch (t) 1 ɛ(t) de(t) ρ T (h pf (t)), ɛ(t) c V d (t) 11 / 16

35 Elements of the event-trigger Recall performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 Channel trigger function: h ch (t) Lemma ɛ(t) de(t) ρ T (h pf (t)), ɛ(t) c V d (t) If h pf (t) 1 and h ch (t) 1 then h pf (s) 1, s [t, t + T ]. 11 / 16

36 Elements of the event-trigger Recall performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 Channel trigger function: h ch (t) Lemma ɛ(t) de(t) ρ T (h pf (t)), ɛ(t) c V d (t) If h pf (t) 1 and h ch (t) 1 then h pf (s) 1, s [t, t + T ]. Idea for triggering: Make sure h pf (t) 1, t [t k, r k ] 11 / 16

37 Elements of the event-trigger Recall performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 Channel trigger function: h ch (t) Lemma ɛ(t) de(t) ρ T (h pf (t)), ɛ(t) c V d (t) If h pf (t) 1 and h ch (t) 1 then h pf (s) 1, s [t, t + T ]. Idea for triggering: Make sure h pf (t) 1, t [t k, r k ] Make sure h ch ( r k ) 1 so that future ability to control is not lost 11 / 16

38 Elements of the event-trigger Recall performance objective: ensure h pf (t) V (x(t)) V d (t) 1, for all t t 0 Channel trigger function: h ch (t) Lemma ɛ(t) de(t) ρ T (h pf (t)), ɛ(t) c V d (t) If h pf (t) 1 and h ch (t) 1 then h pf (s) 1, s [t, t + T ]. Idea for triggering: Make sure h pf (t) 1, t [t k, r k ] Make sure h ch ( r k ) 1 so that future ability to control is not lost L 1 (t) h pf (T (t), h pf (t), ɛ(t)) L 2 (t) h ch (T (t), h pf (t), ɛ(t), ψ τ l(t)) { T (t) TM (ψ τ l(t)), if ψ τ l(t) 1 2 R(t), if ψτ l(t) = / 16

39 Role data capacity in control 12 / 16

40 Role data capacity in control ( ) L 3 (t) n log e µ(τ l (t) t) ɛ(t) 2 ɛ r(t) σ 1 ˆDs (t, τ l (t)) 12 / 16

41 Role data capacity in control R ( ) L 3 (t) n log e µ(τ l (t) t) ɛ(t) 2 ɛ r(t) σ 1 ˆDs (t, τ l (t)) t Transmission policy should be in tune with the optimal allocation 12 / 16

42 Role data capacity in control R ( ) L 3 (t) n log e µ(τ l (t) t) ɛ(t) 2 ɛ r(t) σ 1 ˆDs (t, τ l (t)) t Transmission policy should be in tune with the optimal allocation Φ τ l(t) [ P j R j (t θ j ) ] +, t (θ j, θ j+1 ] (optim. alloc. in (t, θ j+1 ]) 12 / 16

43 Role data capacity in control R ( ) L 3 (t) n log e µ(τ l (t) t) ɛ(t) 2 ɛ r(t) σ 1 ˆDs (t, τ l (t)) t Transmission policy should be in tune with the optimal allocation Φ τ l(t) [ P j R j (t θ j ) ] +, t (θ j, θ j+1 ] (optim. alloc. in (t, θ j+1 ]) Artificial bound on packet size: ψ τ l(t) min{ p(t), Φ τ l(t)} 12 / 16

44 Role data capacity in control R ( ) L 3 (t) n log e µ(τ l (t) t) ɛ(t) 2 ɛ r(t) σ 1 ˆDs (t, τ l (t)) t Transmission policy should be in tune with the optimal allocation Φ τ l(t) [ P j R j (t θ j ) ] +, t (θ j, θ j+1 ] (optim. alloc. in (t, θ j+1 ]) Artificial bound on packet size: ψ τ l(t) min{ p(t), Φ τ l(t)} If L 3 (t k ) 0 and p k ψ τ l(t k ) If data capacity was sufficient at t k and p k respects artificial bound 12 / 16

45 Role data capacity in control R ( ) L 3 (t) n log e µ(τ l (t) t) ɛ(t) 2 ɛ r(t) σ 1 ˆDs (t, τ l (t)) t Transmission policy should be in tune with the optimal allocation Φ τ l(t) [ P j R j (t θ j ) ] +, t (θ j, θ j+1 ] (optim. alloc. in (t, θ j+1 ]) Artificial bound on packet size: ψ τ l(t) min{ p(t), Φ τ l(t)} If L 3 (t k ) 0 and p k ψ τ l(t k ) then L 3 (r k ) 0 If data capacity was sufficient at t k and p k respects artificial bound then data capacity is sufficient at r k 12 / 16

46 Role data capacity in control R ( ) L 3 (t) n log e µ(τ l (t) t) ɛ(t) 2 ɛ r(t) σ 1 ˆDs (t, τ l (t)) t Transmission policy should be in tune with the optimal allocation Φ τ l(t) [ P j R j (t θ j ) ] +, t (θ j, θ j+1 ] (optim. alloc. in (t, θ j+1 ]) Artificial bound on packet size: ψ τ l(t) min{ p(t), Φ τ l(t)} If L 3 (t k ) 0 and p k ψ τ l(t k ) then L 3 (r k ) 0 If data capacity was sufficient at t k and p k respects artificial bound then data capacity is sufficient at r k But ψ τ l(t) can be 0 when p(t) > 0 (artificial blackouts) 12 / 16

47 Control policy in the presence of blackouts { t k+1 = min t r k : ψ τ l (t) 1 ( max{ L 1 (t), L 1 (t + ), L 2 (t), L 2 (t + )} 1 max{ L 3 (t), L )} 3 (t + )} 0, 13 / 16

48 Control policy in the presence of blackouts { t k+1 = min t r k : ψ τ l (t) 1 ( max{ L 1 (t), L 1 (t + ), L 2 (t), L 2 (t + )} 1 max{ L 3 (t), L )} 3 (t + )} 0, p k Z >0 [p k, ψ τ l (t k )] p k min{p Z >0 : h ch (T M (p), h pf (t k ), ɛ(t k ), p) 1}. 13 / 16

49 Control policy in the presence of blackouts { t k+1 = min t r k : ψ τ l (t) 1 ( max{ L 1 (t), L 1 (t + ), L 2 (t), L 2 (t + )} 1 max{ L 3 (t), L )} 3 (t + )} 0, p k Z >0 [p k, ψ τ l (t k )] p k min{p Z >0 : h ch (T M (p), h pf (t k ), ɛ(t k ), p) 1}. r k = min{t r k : ψ τ l (t) 1 p(t) = 0}. 13 / 16

50 Control policy in the presence of blackouts Theorem If R(t) (p+2) T M (p), p {1,..., pmax }, t L1 (t 0 ) 1, L 2 (t 0 ) 1 and L 3 (t 0 ) 0 (initial feasibility) Conditions on blackout lengths 14 / 16

51 Control policy in the presence of blackouts Theorem If R(t) (p+2) T M (p), p {1,..., pmax }, t L1 (t 0 ) 1, L 2 (t 0 ) 1 and L 3 (t 0 ) 0 (initial feasibility) Conditions on blackout lengths Then {t k }, {p k }, { r k } well defined inter-transmission times have uniform positive lower bound V (x(t)) V d (t 0 )e β(t t 0) for t t 0 (origin is exponentially stable) 14 / 16

52 Simulation results: 2D linear system # bits transmitted t (seconds) Channel comm. rate 1.4 x t (seconds) V d Total # bits transmitted t (seconds) t (seconds) 15 / 16

53 Summary Contribution: Fusion of event-triggered control and information-theoretic control 16 / 16

54 Summary Contribution: Fusion of event-triggered control and information-theoretic control Definition and computation of data capacity under full channel information 16 / 16

55 Summary Contribution: Fusion of event-triggered control and information-theoretic control Definition and computation of data capacity under full channel information Control under time-varying channels (including blackouts) Stabilization with prescribed convergence rate 16 / 16

56 Summary Contribution: Fusion of event-triggered control and information-theoretic control Definition and computation of data capacity under full channel information Control under time-varying channels (including blackouts) Stabilization with prescribed convergence rate Future work: Address conservatism in the design Stochastic model of channel evolution Impact of the available information pattern at the encoder 16 / 16