Event-triggered stabilization of linear systems under channel blackouts Pavankumar Tallapragada, Massimo Franceschetti & Jorge Cort´ es Allerton Conference, 30 Sept. 2015 Acknowledgements: National Science Foundation (Grants CNS-1329619, CNS-1446891) 1 / 16
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
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
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
System description Plant dynamics: x ( t ) ∈ R n x ( t ) = Ax ( t ) + Bu ( t ), ˙ u ( t ) = K ˆ x ( t ), 3 / 16
System description Plant dynamics: x ( t ) ∈ R n x ( t ) = Ax ( t ) + Bu ( t ), ˙ u ( t ) = K ˆ x ( t ), Communication model: b k p k ∆ k ≤ ∆( t k , p k ) � R a ( t 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
System description Plant dynamics: x ( t ) ∈ R n x ( t ) = Ax ( t ) + Bu ( t ), ˙ u ( t ) = K ˆ x ( t ), Communication model: b k p k ∆ k ≤ ∆( t k , p k ) � R a ( t 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: ˙ x ( t ) + Bu ( t ) = ¯ x ( t ) = A ˆ ˆ A ˆ x ( t ) , t ∈ [˜ r k , ˜ r k +1 ) 3 / 16
System description Plant dynamics: x ( t ) ∈ R n x ( t ) = Ax ( t ) + Bu ( t ), ˙ u ( t ) = K ˆ x ( t ), Communication model: b k p k ∆ k ≤ ∆( t k , p k ) � R a ( t 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: ˙ x ( t ) + Bu ( t ) = ¯ x ( t ) = A ˆ ˆ A ˆ x ( t ) , t ∈ [˜ r k , ˜ r k +1 ) x ( t − r k ) � q k ( x ( t k ) , ˆ Dynamic controller jump: ˆ x (˜ k )) Encoding error: x e � x − ˆ x 3 / 16
Quantization Can design 2 consistent algorithms for the encoder and decoder to implement quantizer q k so that: 2 Tallapragada, Cort´ es (2016) 4 / 16
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, Cort´ es (2016) 4 / 16
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 1 δ k +1 = 2 p k +1 d e ( t k +1 ) . 2 Tallapragada, Cort´ es (2016) 4 / 16
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 1 δ k +1 = 2 p k +1 d e ( t k +1 ) . • Then, � x e ( t ) � ∞ ≤ d e ( t ), for all t ≥ t 0 2 Tallapragada, Cort´ es (2016) 4 / 16
Objective A T P = − Q Suppose ¯ ⇒ P ¯ A + ¯ A = A + BK is Hurwitz ⇐ Lyapunov function: x �→ V ( x ) = x T Px 5 / 16
Objective A T P = − Q Suppose ¯ ⇒ P ¯ A + ¯ A = A + BK is Hurwitz ⇐ Lyapunov function: x �→ V ( x ) = x T Px 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
Objective A T P = − Q Suppose ¯ ⇒ P ¯ A + ¯ A = A + BK is Hurwitz ⇐ Lyapunov function: x �→ V ( x ) = x T Px 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
Time-slotted channel model 3500 10 8 3000 6 ¯ p R 2500 4 2 2000 0 0 2 4 6 8 10 0 2 4 6 8 10 t t p k ∀ t ∈ ( θ j , θ j +1 ] , ∆( t k , p k ) ≥ R ( t k ) R ( t ) = R j , min comm. rate: p ( t ) = ¯ ¯ ∀ t ∈ ( θ j , θ j +1 ] , max packet size: p k ≤ ¯ p ( t k ) π j , • 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
Time-slotted channel model 3500 10 8 3000 6 ¯ p R 2500 4 2 2000 0 0 2 4 6 8 10 0 2 4 6 8 10 t t p k ∀ t ∈ ( θ j , θ j +1 ] , ∆( t k , p k ) ≥ R ( t k ) R ( t ) = R j , min comm. rate: p ( t ) = ¯ ¯ ∀ t ∈ ( θ j , θ j +1 ] , max packet size: p k ≤ ¯ p ( t k ) π j , • 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
Time-slotted channel model 3500 10 8 3000 6 ¯ p R 2500 4 2 2000 0 0 2 4 6 8 10 0 2 4 6 8 10 t t p k ∀ t ∈ ( θ j , θ j +1 ] , ∆( t k , p k ) ≥ R ( t k ) R ( t ) = R j , min comm. rate: p ( t ) = ¯ ¯ ∀ t ∈ ( θ j , θ j +1 ] , max packet size: p k ≤ ¯ p ( t k ) π j , • 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
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 } k τ 2 � D ( τ 1 , τ 2 ) � max n p k { t k } , { p k } k τ 1 = 3, k τ 2 = 7 k = k τ 1 s.t. . . . 7 / 16
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 } k τ 2 � D ( τ 1 , τ 2 ) � max n p k { t k } , { p k } k τ 1 = 3, k τ 2 = 7 k = k τ 1 s.t. . . . Equivalent to optimal allocation of discrete # bits to be transmitted in each time slot 7 / 16
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 10 8 6 ¯ p 4 2 0 0 2 4 6 8 10 t 8 / 16
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 � T j ( φ j f T j ( φ j f nR j ¯ if ¯ j 0 ) + n ¯ π j , j 0 ) > 0 nφ j ≤ 0 otherwise 10 8 6 ¯ p 4 2 0 0 2 4 6 8 10 t 8 / 16
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 � T j ( φ j f T j ( φ j f nR j ¯ if ¯ j 0 ) + n ¯ π j , j 0 ) > 0 nφ j ≤ 0 otherwise Count only the bits also received 10 � ¯ 8 T j ( φ j f T j ( φ j f if ¯ j 0 ) + θ j f − θ j +1 , j 0 ) > 0 6 φ j ¯ p R j ≤ 4 2 0 , otherwise . 0 0 2 4 6 8 10 t 8 / 16
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 � T j ( φ j f T j ( φ j f nR j ¯ if ¯ j 0 ) + n ¯ π j , j 0 ) > 0 nφ j ≤ 0 otherwise Count only the bits also received 10 � ¯ 8 T j ( φ j f T j ( φ j f if ¯ j 0 ) + θ j f − θ j +1 , j 0 ) > 0 6 φ j ¯ p R j ≤ 4 2 0 , otherwise . 0 0 2 4 6 8 10 t j f − 1 � D ( θ j 0 , θ j f ) = max n φ j . φ j ∈ Z ≥ 0 j = j 0 s.t. . . . 8 / 16
A suboptimal solution for “slowly varying channels” Proposition Assume ¯ π j < T j +1 , ∀ j ∈ N j f j 0 (any bits transmitted in slot j are R j received before the end of slot j + 1 ). 9 / 16
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