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A New Perspective on Quality Evaluation for Control Systems with Stochastic Timing Maximilian Gaukler , Andreas Michalka, Peter Ulbrich and Tobias Klaus April 11th, 2018 Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 1

Motivation Disturbance Measurement noise Quality of Control : How well does the control system Plant work under • random disturbance Input/Output Timing • I/O timing ? Time-varying situation : • execution conditions Controller • disturbance amplitude • reference trajectory Other Applications and Controllers � Quality is time-varying Real-Time Computing System Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation Disturbance Measurement noise Quality of Control : How well does the control system Plant work under • random disturbance Input/Output Timing • I/O timing ? Time-varying situation : • execution conditions Controller • disturbance amplitude • reference trajectory Other Applications and Controllers � Quality is time-varying Real-Time Computing System Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation Disturbance Measurement noise Quality of Control : How well does the control system Plant work under • random disturbance Input/Output Timing • I/O timing ? Time-varying situation : • execution conditions Controller • disturbance amplitude • reference trajectory Other Applications and Controllers � Quality is time-varying Real-Time Computing System Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation Disturbance Measurement noise Quality of Control : How well does the control system Plant work under • random disturbance Input/Output Timing • I/O timing ? Time-varying situation : • execution conditions Controller • disturbance amplitude • reference trajectory Other Applications and Controllers � Quality is time-varying Real-Time Computing System Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation Disturbance Measurement noise Quality of Control : How well does the control system Plant work under • random disturbance Input/Output Timing • I/O timing ? Time-varying situation : • execution conditions Controller • disturbance amplitude • reference trajectory Other Applications and Controllers � Quality is time-varying Real-Time Computing System Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation Disturbance Measurement noise Quality of Control : How well does the control system Plant work under • random disturbance Input/Output Timing • I/O timing ? Time-varying situation : • execution conditions Controller • disturbance amplitude • reference trajectory Other Applications and Controllers � Quality is time-varying Real-Time Computing System Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Related Work and Topics Sampled-Data Control with uncertain timing Co-Design Analysis � necessary? worst-case guarantee typical performance (deterministic) (stochastic) time-averaged (stationary) time-varying JITTERBUG • simulation (Lincoln and Cervin 2002) slow, no formal insight • efficient computation? � aim of this work Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 3

Contents 1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 4

1 Problem Formulation 1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 5

1 Problem Formulation • continuous-time MIMO plant (linear, time-invariant) x p ( t ) = A p x p ( t ) + B p u ( t ) + G p d ( t ) , ˙ x p (0) = 0 , y ( t ) = C p x p ( t ) + w p ( t ) • d ( t ) : stochastic disturbance (white noise, time-varying covariance H ( t ) ) • w p ( t ) : measurement noise • discrete-time controller (linear + reference trajectory), sampling time T x d [ k + 1] = A d [ k ] x d [ k ] + B d [ k ] y [ k ] + f d [ k ] , u [ k ] = C d [ k ] x d [ k ] + g d [ k ] , x d [0] = 0 • sampling and actuation delays | ∆ t ... | < T/ 2 • time-varying, per sensor/actuator component • random, independent of disturbance and measurement noise Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation • continuous-time MIMO plant (linear, time-invariant) x p ( t ) = A p x p ( t ) + B p u ( t ) + G p d ( t ) , ˙ x p (0) = 0 , y ( t ) = C p x p ( t ) + w p ( t ) • d ( t ) : stochastic disturbance (white noise, time-varying covariance H ( t ) ) • w p ( t ) : measurement noise • discrete-time controller (linear + reference trajectory), sampling time T x d [ k + 1] = A d [ k ] x d [ k ] + B d [ k ] y [ k ] + f d [ k ] , u [ k ] = C d [ k ] x d [ k ] + g d [ k ] , x d [0] = 0 • sampling and actuation delays | ∆ t ... | < T/ 2 • time-varying, per sensor/actuator component • random, independent of disturbance and measurement noise Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation • continuous-time MIMO plant (linear, time-invariant) x p ( t ) = A p x p ( t ) + B p u ( t ) + G p d ( t ) , ˙ x p (0) = 0 , y ( t ) = C p x p ( t ) + w p ( t ) • d ( t ) : stochastic disturbance (white noise, time-varying covariance H ( t ) ) • w p ( t ) : measurement noise • discrete-time controller (linear + reference trajectory), sampling time T x d [ k + 1] = A d [ k ] x d [ k ] + B d [ k ] y [ k ] + f d [ k ] , u [ k ] = C d [ k ] x d [ k ] + g d [ k ] , x d [0] = 0 • sampling and actuation delays | ∆ t ... | < T/ 2 • time-varying, per sensor/actuator component • random, independent of disturbance and measurement noise Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation • continuous-time MIMO plant (linear, time-invariant) x p ( t ) = A p x p ( t ) + B p u ( t ) + G p d ( t ) , ˙ x p (0) = 0 , y ( t ) = C p x p ( t ) + w p ( t ) • d ( t ) : stochastic disturbance (white noise, time-varying covariance H ( t ) ) • w p ( t ) : measurement noise • discrete-time controller (linear + reference trajectory), sampling time T x d [ k + 1] = A d [ k ] x d [ k ] + B d [ k ] y [ k ] + f d [ k ] , u [ k ] = C d [ k ] x d [ k ] + g d [ k ] , x d [0] = 0 • sampling and actuation delays | ∆ t ... | < T/ 2 • time-varying, per sensor/actuator component • random, independent of disturbance and measurement noise Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation • continuous-time MIMO plant (linear, time-invariant) x p ( t ) = A p x p ( t ) + B p u ( t ) + G p d ( t ) , ˙ x p (0) = 0 , y ( t ) = C p x p ( t ) + w p ( t ) • d ( t ) : stochastic disturbance (white noise, time-varying covariance H ( t ) ) • w p ( t ) : measurement noise • discrete-time controller (linear + reference trajectory), sampling time T x d [ k + 1] = A d [ k ] x d [ k ] + B d [ k ] y [ k ] + f d [ k ] , u [ k ] = C d [ k ] x d [ k ] + g d [ k ] , x d [0] = 0 • sampling and actuation delays | ∆ t ... | < T/ 2 • time-varying, per sensor/actuator component • random, independent of disturbance and measurement noise Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation • continuous-time MIMO plant (linear, time-invariant) x p ( t ) = A p x p ( t ) + B p u ( t ) + G p d ( t ) , ˙ x p (0) = 0 , y ( t ) = C p x p ( t ) + w p ( t ) • d ( t ) : stochastic disturbance (white noise, time-varying covariance H ( t ) ) • w p ( t ) : measurement noise • discrete-time controller (linear + reference trajectory), sampling time T x d [ k + 1] = A d [ k ] x d [ k ] + B d [ k ] y [ k ] + f d [ k ] , u [ k ] = C d [ k ] x d [ k ] + g d [ k ] , x d [0] = 0 • sampling and actuation delays | ∆ t ... | < T/ 2 • time-varying, per sensor/actuator component • random, independent of disturbance and measurement noise Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation • quality via quadratic cost function: deviation from reference x r , u r J ( t ) =( x p ( t ) − x r ( t )) T ˜ Q ( x p ( t ) − x r ( t )) + ( u ( t ) − u r ( t )) T ˜ R ( u ( t ) − u r ( t )) with x r ( t ) , u r ( t ) known a priori. • desired result: expected cost E{ J ( t ) } (time-varying) Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 7

1 Problem Formulation • quality via quadratic cost function: deviation from reference x r , u r J ( t ) =( x p ( t ) − x r ( t )) T ˜ Q ( x p ( t ) − x r ( t )) + ( u ( t ) − u r ( t )) T ˜ R ( u ( t ) − u r ( t )) with x r ( t ) , u r ( t ) known a priori. • desired result: expected cost E{ J ( t ) } (time-varying) Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 7

2 Reformulation as Linear Impulsive System 1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 8