Event-Triggered Control Design with Performance Barrier Pio Ong and - PowerPoint PPT Presentation

Event-Triggered Control Design with Performance Barrier Pio Ong and Jorge Cort´ es Mechanical and Aerospace Engineering University of California, San Diego http://carmenere.ucsd.edu/jorge 57th IEEE Conference on Decision and Control: Robust Event-Triggered Control Miami, Florida December 17-19, 2018

Motivating Example Example from [P. Tabuada 2007], a well-cited paper in ET control � � � � � � � � x 1 ˙ 0 1 x 1 0 = + u x 2 ˙ − 2 3 x 2 1 P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 2 / 17

Motivating Example Example from [P. Tabuada 2007], a well-cited paper in ET control � � � � � � � � x 1 ˙ 0 1 x 1 0 = + u x 2 ˙ − 2 3 x 2 1 Continuous time : Pick u = x 1 − 4 x 2 P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 2 / 17

Motivating Example Example from [P. Tabuada 2007], a well-cited paper in ET control � � � � � � � � x 1 ˙ 0 1 x 1 0 = + u x 2 ˙ − 2 3 x 2 1 Continuous time : Pick u = x 1 − 4 x 2 Event-triggered : Update u above when � e � = σ � x � , e = x − x k σ is a design parameter. Lower → better performance Higher → less trigger, conserving resources P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 2 / 17

Key Idea to Take Away from My Talk Motivation: Unclear on how to tune the design parameter to create a balance between 1 trigger frequency and performance Standard ET design scheme can be inefficient in achieving desired performance 2 Assumption: Desired performance can be achieved in continuous time 1 Approach: ˙ Throw away the Lyapunov’s criterion for stability, i.e. V ≤ 0 1 Allow ˙ V > 0 2 Incorporate performance requirement into the trigger condition 3 P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 3 / 17

Outline Event-triggered control design overview: Linear system example Identify inefficiencies in satisfying a given desired performance Our design: Incorporating performance requirement Use barrier concept Advantages Apply our new design idea to distributed cases Wrapping Up My Talk Simulations Conclusion and future ideas P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 4 / 17

Design Parameter How might σ be picked? From earlier example: Lyapunov function � 1 � 1 / 4 V ≤ − 0 . 44 � x � 2 + 8 � e �� x � ˙ V ( x ) = x T x = ⇒ 1 / 4 1 using ET control, ˙ V ≤ − (0 . 44 − 8 σ ) � x 2 � , σ = 0 . 05 was picked, but why? P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 5 / 17

Design Parameter How might σ be picked? From earlier example: Lyapunov function � 1 � 1 / 4 V ≤ − 0 . 44 � x � 2 + 8 � e �� x � ˙ V ( x ) = x T x = ⇒ 1 / 4 1 using ET control, ˙ V ≤ − (0 . 44 − 8 σ ) � x 2 � , σ = 0 . 05 was picked, but why? Maybe because this σ guarantees the performance of V ≤ − 0 . 04 ˙ 3 / 4 V = > V ( x ( t )) ≤ V ( x 0 ) exp( − 0 . 032 t ) P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 5 / 17

Design Parameter How might σ be picked? From earlier example: Lyapunov function � 1 � 1 / 4 V ≤ − 0 . 44 � x � 2 + 8 � e �� x � ˙ V ( x ) = x T x = ⇒ 1 / 4 1 using ET control, ˙ V ≤ − (0 . 44 − 8 σ ) � x 2 � , σ = 0 . 05 was picked, but why? Maybe because this σ guarantees the performance of V ≤ − 0 . 04 ˙ 3 / 4 V = > V ( x ( t )) ≤ V ( x 0 ) exp( − 0 . 032 t ) We can reverse the process. Given performance specification S ( t ) ≤ V ( x 0 ) exp( − rt ), one can find σ P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 5 / 17

Assumptions made Assumptions so that we can reverse the process P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 6 / 17

Assumptions made Assumptions so that we can reverse the process Known ISS Lyapunov function (same [P. Tabuada 2007]) ˙ V ≤ − α ( � x � ) + γ ( � e � ) P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 6 / 17

Assumptions made Assumptions so that we can reverse the process Known ISS Lyapunov function (same [P. Tabuada 2007]) ˙ V ≤ − α ( � x � ) + γ ( � e � ) Given specification function ˙ S = − h ( S ) , S ( x 0 , 0) ≥ V ( x 0 ) where h locally Lipschitz, class K Note: earlier, special case ˙ S = − rS P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 6 / 17

Assumptions made Assumptions so that we can reverse the process Known ISS Lyapunov function (same [P. Tabuada 2007]) ˙ V ≤ − α ( � x � ) + γ ( � e � ) Given specification function ˙ S = − h ( S ) , S ( x 0 , 0) ≥ V ( x 0 ) where h locally Lipschitz, class K Note: earlier, special case ˙ S = − rS Performance achievable in continuous time − α ( � x � ) < − h ( ¯ V ( x )) P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 6 / 17

Derivative based Trigger Design idea : forget σ , make ˙ V ≤ − h ( V ), then V ≤ S (Comparison Lemma) Derivative-based ET � � t > t k | g ( x ( t ) , e ( t )) + h ( ¯ t k +1 = min V ( x ( t ))) = 0 t where L f V ( x ) ≤ g ( x , e ) ≤ − α ( � x � ) + γ ( � e � ) P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 7 / 17

Derivative based Trigger Design idea : forget σ , make ˙ V ≤ − h ( V ), then V ≤ S (Comparison Lemma) Derivative-based ET � � t > t k | g ( x ( t ) , e ( t )) + h ( ¯ t k +1 = min V ( x ( t ))) = 0 t where L f V ( x ) ≤ g ( x , e ) ≤ − α ( � x � ) + γ ( � e � ) Ex. update u when − 0 . 44 � x � 2 + 8 � e �� x � + 0 . 032 V ( x ) = 0 P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 7 / 17

Derivative based Trigger Design idea : forget σ , make ˙ V ≤ − h ( V ), then V ≤ S (Comparison Lemma) Derivative-based ET � � t > t k | g ( x ( t ) , e ( t )) + h ( ¯ t k +1 = min V ( x ( t ))) = 0 t where L f V ( x ) ≤ g ( x , e ) ≤ − α ( � x � ) + γ ( � e � ) Ex. update u when − 0 . 44 � x � 2 + 8 � e �� x � + 0 . 032 V ( x ) = 0 100 Desired Bound Derivative-based 80 Lyapunov Function 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 time (sec) 0 1 2 3 4 5 6 7 8 9 10 P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 7 / 17

Derivative based Trigger Design idea : forget σ , make ˙ V ≤ − h ( V ), then V ≤ S (Comparison Lemma) Derivative-based ET � � t > t k | g ( x ( t ) , e ( t )) + h ( ¯ t k +1 = min V ( x ( t ))) = 0 t where L f V ( x ) ≤ g ( x , e ) ≤ − α ( � x � ) + γ ( � e � ) Ex. update u when − 0 . 44 � x � 2 + 8 � e �� x � + 0 . 032 V ( x ) = 0 100 Desired Bound Derivative-based 80 Lyapunov Function 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 time (sec) 0 1 2 3 4 5 6 7 8 9 10 Problem? The trigger is too early. There is room for improvements. P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 7 / 17

Lyapunov Function Trigger Design idea : just make V ≤ S Function-based ET � � t k +1 = t > t k | S ( x 0 , t ) − V ( x ( t )) = 0 P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 8 / 17

Lyapunov Function Trigger Design idea : just make V ≤ S Function-based ET � � t k +1 = t > t k | S ( x 0 , t ) − V ( x ( t )) = 0 Straightforward. Performance immediately satisfied 100 Desired Bound Function-based 80 Lyapunov Function 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 time (sec) 0 1 2 3 4 5 6 7 8 9 10 Efficient, less triggers, but there is no robustness to time delay P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 8 / 17

Performance Barrier Design Design idea : combine the two schemes, but how? P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 9 / 17

Performance Barrier Design Design idea : combine the two schemes, but how? Answer: Use the concept from control barrier function P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 9 / 17

Performance Barrier Design Design idea : combine the two schemes, but how? Answer: Use the concept from control barrier function Performance Barrier ET � � �� t > t k | g ( x ( t ) , e ( t )) + h ( ¯ S ( x 0 , t ) − ¯ t k +1 = min V ( x ( t ))) = β V ( x ( t )) t � �� � derivative − based where β is class- K ∞ P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 9 / 17

Performance Barrier Design Design idea : combine the two schemes, but how? Answer: Use the concept from control barrier function Performance Barrier ET � � �� t > t k | g ( x ( t ) , e ( t )) + h ( ¯ S ( x 0 , t ) − ¯ t k +1 = min V ( x ( t ))) = β V ( x ( t )) t � �� � derivative − based where β is class- K ∞ What have we done here? We allow ˙ V > − h ( V ) given some performance “residual”, S − V > 0 P. Ong (UCSD) Event-Triggered Control w/ Performance Barrier December 17, 2018 9 / 17