Event-triggered stabilization of linear systems under bounded bit rates Pavankumar Tallapragada & Jorge Cort´ es Department of Mechanical and Aerospace Engineering Conference on Decision and Control, 15 December 2014 Acknowledgements: National Science Foundation (Grant CNS-1329619) 1 / 18
Networked control systems 2 / 18
Networked control systems • When to transmit: Event-triggered strategies • A trigger function encodes the control goal • Transmissions occur only when necessary • Better use of resources than time-triggered 2 / 18
Networked control systems • What to transmit: Information-theory based data rate theorems • Quite successful in the discrete-time setting • Tight necessary and sufficient data rates are available 3 / 18
Unanswered questions Event-triggered control: • What is the average inter-tx time? 4 / 18
Unanswered questions Event-triggered control: • What is the average inter-tx time? • More generally, what is the average data rate? 4 / 18
Unanswered questions Event-triggered control: • What is the average inter-tx time? • More generally, what is the average data rate? • Given a bound on the channel capacity, what should the transmission policy be? 4 / 18
Unanswered questions Event-triggered control: • What is the average inter-tx time? • More generally, what is the average data rate? • Given a bound on the channel capacity, what should the transmission policy be? Information-theoretic control: • There is still a lot of scope for work in the continuous-time setting • How to design controllers with specified performance (e.g. convergence rate)? 4 / 18
Unanswered questions Event-triggered control: • What is the average inter-tx time? • More generally, what is the average data rate? • Given a bound on the channel capacity, what should the transmission policy be? Information-theoretic control: • There is still a lot of scope for work in the continuous-time setting • How to design controllers with specified performance (e.g. convergence rate)? The two themes have complementary strengths 4 / 18
System description Plant dynamics: x ( t ) = Ax ( t ) + Bu ( t ) + v ( t ), ˙ u ( t ) = K ˆ x ( t ) � v ( t ) � 2 ≤ ν, ∀ t ∈ [0 , ∞ ) 5 / 18
System description Plant dynamics: x ( t ) = Ax ( t ) + Bu ( t ) + v ( t ), ˙ u ( t ) = K ˆ x ( t ) � v ( t ) � 2 ≤ ν, ∀ t ∈ [0 , ∞ ) Transmission times: { t k } k ∈ N , Reception times: { r k } k ∈ N ∆ k � r k − t k = ∆( t k , p k ), np k is the number of bits transmitted at t k 5 / 18
System description Plant dynamics: x ( t ) = Ax ( t ) + Bu ( t ) + v ( t ), ˙ u ( t ) = K ˆ x ( t ) � v ( t ) � 2 ≤ ν, ∀ t ∈ [0 , ∞ ) Transmission times: { t k } k ∈ N , Reception times: { r k } k ∈ N ∆ k � r k − t k = ∆( t k , p k ), np k is the number of bits transmitted at t k Dynamic controller flow: ˙ x ( t ) + Bu ( t ) = ¯ x ( t ) = A ˆ ˆ A ˆ x ( t ) , t ∈ [ r k , r k +1 ) 5 / 18
System description Plant dynamics: x ( t ) = Ax ( t ) + Bu ( t ) + v ( t ), ˙ u ( t ) = K ˆ x ( t ) � v ( t ) � 2 ≤ ν, ∀ t ∈ [0 , ∞ ) Transmission times: { t k } k ∈ N , Reception times: { r k } k ∈ N ∆ k � r k − t k = ∆( t k , p k ), np k is the number of bits transmitted at t 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 − x ( r k ) � q k ( x ( t k ) , ˆ Dynamic controller jump: ˆ k )) 5 / 18
System description Plant dynamics: x ( t ) = Ax ( t ) + Bu ( t ) + v ( t ), ˙ u ( t ) = K ˆ x ( t ) � v ( t ) � 2 ≤ ν, ∀ t ∈ [0 , ∞ ) Transmission times: { t k } k ∈ N , Reception times: { r k } k ∈ N ∆ k � r k − t k = ∆( t k , p k ), np k is the number of bits transmitted at t 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 − x ( r k ) � q k ( x ( t k ) , ˆ Dynamic controller jump: ˆ k )) Closed loop flow, for t ∈ [ r k , r k +1 ) x ( t ) = ¯ ¯ A � A + BK ˙ Ax ( t ) − BKx e ( t ) + v ( t ) , x e � x − ˆ x e ( t ) = Ax e ( t ) + v ( t ) , ˙ x (encoding error) 5 / 18
Quantization and coding (instant communication) If the decoder knows d e ( t 0 ) s.t. � x e ( t 0 ) � ∞ ≤ d e ( t 0 ) 6 / 18
Quantization and coding (instant communication) If the decoder knows d e ( t 0 ) s.t. � x e ( t 0 ) � ∞ ≤ d e ( t 0 ) Both encoder and decoder compute recursively: ν [ e � A � 2 ( t − t k ) − 1] , t ∈ [ t k , t k +1 ) d e ( t ) � � e A ( t − t k ) � ∞ d e ( t k ) + � A � 2 1 2 p k +1 d e ( t − d e ( t k +1 ) = k +1 ) 6 / 18
Quantization and coding (instant communication) If the decoder knows d e ( t 0 ) s.t. � x e ( t 0 ) � ∞ ≤ d e ( t 0 ) Both encoder and decoder compute recursively: ν [ e � A � 2 ( t − t k ) − 1] , t ∈ [ t k , t k +1 ) d e ( t ) � � e A ( t − t k ) � ∞ d e ( t k ) + � A � 2 1 2 p k +1 d e ( t − d e ( t k +1 ) = k +1 ) Then, � x e ( t ) � ∞ ≤ d e ( t ), for all t ≥ t 0 d e ( t − k ) defines the quantization domain at time t k # bits used to quantize at time t k is np k 6 / 18
Quantization and coding (instant communication) If the decoder knows d e ( t 0 ) s.t. � x e ( t 0 ) � ∞ ≤ d e ( t 0 ) Both encoder and decoder compute recursively: ν [ e � A � 2 ( t − t k ) − 1] , t ∈ [ t k , t k +1 ) d e ( t ) � � e A ( t − t k ) � ∞ d e ( t k ) + � A � 2 1 2 p k +1 d e ( t − d e ( t k +1 ) = k +1 ) Then, � x e ( t ) � ∞ ≤ d e ( t ), for all t ≥ t 0 d e ( t − k ) defines the quantization domain at time t k # bits used to quantize at time t k is np k Non-instant communication: more involved 6 / 18
Control objective A T P = − Q Suppose ¯ ⇒ P ¯ A + ¯ A = A + BK is Hurwitz ⇐ Lyapunov function: x �→ V ( x ) = x T Px 7 / 18
Control 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 ) − V 0 ) e − β ( t − t 0 ) + V 0 Performance objective: ensure b ( t ) � V ( x ( t )) V d ( t ) ≤ 1, for all t ≥ t 0 7 / 18
Control 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 ) − V 0 ) e − β ( t − t 0 ) + V 0 Performance objective: ensure b ( t ) � V ( x ( t )) V d ( t ) ≤ 1, for all t ≥ t 0 Design objective: • Design event-triggered communication policy that recursively determines { t k } and np k • Ensure a uniform positive lower bound for { t k − t k − 1 } k ∈ N • Ensure np k is upper bounded by the given “channel capacity” • Quantify the average data rate 7 / 18
Necessary data rate (non-state-triggered transmissions) Set S ( t ) must lie within the set V d ( t ) � { ξ ∈ R n : V ( ξ ) ≤ V d ( t ) } at all times. 8 / 18
Necessary data rate (non-state-triggered transmissions) Set S ( t ) must lie within the set V d ( t ) � { ξ ∈ R n : V ( ξ ) ≤ V d ( t ) } at all times. Number of bits necessary to be transmitted between t 0 and t to meet the control goal: � vol( S ( t 0 )) � tr( A ) + nβ � � B ( t, t 0 ) ≥ log 2 ( e )( t − t 0 ) + log 2 n 2 c P ( V d ( t 0 )) 2 B ( t, t 0 ) tr( A ) + nβ R as � lim � � ≥ log 2 ( e ) t − t 0 2 t →∞ Assuming all eigenvalues of A have real parts greater than − β . 8 / 18
Control with arbitrary finite communication rate Theorem Assuming control goal is met with continuous and unquantized feedback, let b ( t ) = V ( x ( t )) � � t ≥ t k : b ( t ) ≥ 1 , ˙ t k +1 = min b ( t ) ≥ 0 , V d ( t ) � � d e ( t − � � k ) np k ≥ np k � n log 2 np k : # bits sent at t k , � V d ( t k ) c Then • Inter-transmission times have a uniform positive lower bound, • V ( x ( t )) ≤ V d ( t ) for all t ≥ t 0 9 / 18
Control with arbitrary finite communication rate Theorem Assuming control goal is met with continuous and unquantized feedback, let b ( t ) = V ( x ( t )) � � t ≥ t k : b ( t ) ≥ 1 , ˙ t k +1 = min b ( t ) ≥ 0 , V d ( t ) � � d e ( t − � � k ) np k ≥ np k � n log 2 np k : # bits sent at t k , � V d ( t k ) c Then • Inter-transmission times have a uniform positive lower bound, • V ( x ( t )) ≤ V d ( t ) for all t ≥ t 0 No uniform bound on p k : for special initial conditions p k can be arbitrarily large 9 / 18
Upper bound on the sufficient data rate Corollary If no disturbances, then for any k ∈ N , � � � � d e ( t 0 ) n ( p k + � k − 1 � A � ∞ + β c √ i =1 p i ) ≤ n log 2 ( e )( t k − t 0 )+ n log 2 + n. 2 V d ( t 0 ) • Linear dependence on t k − t 0 • Similar to the necessary data rate (e.g. tr( A ) → n � A � ∞ ) • If more bits than sufficient are transmitted in the past, ( p i > p i for some i < k ), then fewer bits are sufficient at t k • For any k ∈ N , if t k − t k − 1 is bounded, then so is p k • Data rate is bounded even though “communication rate” ( p k ) is not uniformly bounded 10 / 18
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