Feedback Control for Real-Time Systems Chenyang Lu Cyber-Physical - PowerPoint PPT Presentation

Feedback Control for Real-Time Systems Chenyang Lu Cyber-Physical Systems Laboratory Department of Computer Science and Engineering CPS Week 2013

Outline q CPU UClizaCon Control for Distributed Real-Time Systems q Model PredicCve Control q Thermal Control for Real-Time Systems q Nested Control Design CPS Week 2013 2

Outline q CPU UClizaCon Control for Distributed Real-Time Systems q Model PredicCve Control q Thermal Control for Real-Time Systems q Nested Control Design CPS Week 2013 3

Control for Distributed Real-Time Systems q Common characterisCcs of compuCng problems q MIMO: mulC-input (knobs), mulC-output (objecCves) q Coupling between objecCves. q Constraints on knobs. q Model PredicCve Control q OpCmizaCon + PredicCon + Feedback CPS Week 2013 4

Why CPU U?liza?on Control? q Overload protecCon q CPU over-uClizaCon à system crash q Meet response Cme requirement q CPU uClizaCon < bound à meet deadlines CPS Week 2013 5

Challenge: Uncertain?es q ExecuCon Cmes? q Unknown sensor data or user input q Request arrival rate? q Aperiodic events q Bursty service requests q Disturbance? q Denial of Service a[acks Control-theoreCc approach à Robust uClizaCon control in face of workload uncertainty CPS Week 2013 6

End-to-End Tasks in Distributed Systems q Task T i : sequence of subtasks {T ij } on different processors q Periodic: All the subtasks of a task run at a same rate. q Task rate can be adjusted q Within a range q Higher rate à higher uClity T 1 T 11 T 12 T 13 T 3 Remote Invocation T 2 Subtask P 2 P 3 P 1 CPS Week 2013 7

Problem Formula?on q B i : UClizaCon set point of processor P i (1 ≤ i ≤ n) q u i (k): UClizaCon of P i in the k th sampling period q r j (k): Rate of task T j (1 ≤ j ≤ m) in the k th sampling period n ∑ ( B i − u i ( k )) 2 min { r j ( k )|1 ≤ j ≤ n } i = 1 subject to rate constraint: R min,j ≤ r j (k) ≤ R max,j (1 ≤ j ≤ m) CPS Week 2013 8

Single-Input-Single-Output (SISO) Control Single Processor Sensor Inputs {r(k+1)} Set point Application Controller Actuator U s = 69% Middleware u(k) Task Rates R 1 : [1, 5] Hz Monitor OS R 2 : [10, 20] Hz Processor C. Lu, X. Wang, and C. Gill, Feedback Control Real-Time Scheduling in ORB Middleware, IEEE Real-Time and Embedded Technology and Applications Symposium (RTAS'03), May 2003. CPS Week 2013 9

New in Distributed Systems q Need to control uClizaCon of mulCple CPUs q UClizaCon of CPUs are coupled due to end-to-end tasks à ReplicaCng a SISO controller on all processors does not work! q Constraints on task rates T 1 T 11 T 12 T 13 T 3 T 2 P 2 P 3 P 1 CPS Week 2013 10

EUCON: Mul?-Input-Mul?-Output Control u ( k ) ⎡ ⎤ Measured 1 ⎢ ⎥ � Output ⎢ ⎥ Distributed System u ( k ) ⎢ ⎥ ⎣ ⎦ n (m tasks, n processors) UM UM Utilization ! $ ! $ R min,1 R max,1 B Monitor Model # & # & 1 Predictive # & #  & ,   Rate # & Controller # & B n R min, m R max, m Modulator # & RM RM # & " % " % r ( k ) Δ ⎡ ⎤ 1 Feedback Loop Control ⎢ ⎥ � Remote Invocation ⎢ ⎥ Input r ( k ) ⎢ ⎥ Subtask Δ ⎣ ⎦ m C. Lu, X. Wang and X. Koutsoukos, Feedback Utilization Control in Distributed Real-Time Systems with End-to-End Tasks, IEEE Transactions on Parallel and Distributed Systems, 16(6): 550-561, June 2005. CPS Week 2013 11

Control Design Methodology 1. Derive a dynamic model of the system 2. Design a controller 3. Analyze stability CPS Week 2013 12

Dynamic Model: One Processor ∑ u i ( k ) = u i ( k − 1) + g i c jl Δ r j ( k − 1) T jl ∈ S i q S i : set of subtasks on P i q c jl : esCmated execuCon Cme of T il q g i : uClizaCon gain of P i q raCo between actual and esCmated change in uClizaCon q models uncertainty in execuCon Cmes CPS Week 2013 13

Dynamic Model: Mul?ple Processors u ( k ) = u ( k -1) + GF Δ r ( k -1) q G: diagonal matrix of uClizaCon gains q F : subtask allocaCon matrix q models the coupling among processors q f ij = c jl if task T j has a subtask T jl on processor P i q f ij = 0 if T j has no subtask on P i T 1 T 11 c c 0 ⎡ ⎤ T 22 11 21 F = ⎢ ⎥ T 3 0 c c ⎣ ⎦ 22 31 T 2 T 21 T 31 P 1 P 2 CPS Week 2013 14

Model Predic?ve Control q Suitable for coupled MIMO control problems with constraints. q Compute input to minimize cost over a future interval. q Cost funcCon: tracking error and control cost. q Predict cost based on a system model and feedback. q Compute input subject to constraints. q OpCmizaCon + PredicCon + Feedback CPS Week 2013 15

Cost Func?on q Cost P M − 1 2 2 ∑ ∑ V ( k ) = u ( k + i ) − ref ( k + i ) Δ r ( k + i ) − Δ r ( k + i − 1) + i = 1 i = 0 Tracking Error Control Cost q Reference trajectory: exponenCal convergence to B − T s i T ref ref ( k + i ) = B − e ( B − u ( k )) CPS Week 2013 16

Model Predic?ve Controller At the end of each sampling period Compute inputs in future sampling periods q Δ r (k), Δ r (k+1), ... Δ r (k+M-1) to minimize the cost funcCon Cost is predicted using q (1) feedback u(k-1) (2) approximate dynamic model Apply Δ r (k) to the system q At the end of the next sampling period Shic Cme window and re-compute Δ r (k+1), Δ r (k+2), ... Δ r (k+M) based q on feedback CPS Week 2013 17

EUCON Controller Constrained op?miza?on solver B B ⎡ ⎡ ⎤ ⎤ System System Rate Rate 1 1 ⎢ ⎢ ⎥ ⎥ Model Model Constraints Constraints � � ⎢ ⎢ ⎥ ⎥ r r ( ( k k ) ) r ( k 1 ) Δ Δ ⎡ ⎡ ⎤ ⎤ B B Δ + ⎡ ⎤ ⎢ ⎢ ⎥ ⎥ 1 1 ⎣ ⎣ ⎦ ⎦ 1 n n ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ � � Least Squares Solver Least Squares Solver � ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ u u ( ( k k ) ) ⎡ ⎡ ⎤ ⎤ r r ( ( k k ) ) ⎢ ⎢ ⎥ ⎥ Δ Δ r ( k 1 ) 1 1 ⎢ ⎥ Δ + ⎣ ⎣ ⎦ ⎦ m m ⎣ ⎦ m ⎢ ⎢ ⎥ ⎥ � � Cost Cost Reference Reference ⎢ ⎢ ⎥ ⎥ u u ( ( k k ) ) Function Function Trajectory Trajectory ⎢ ⎢ ⎥ ⎥ ⎣ ⎣ ⎦ ⎦ n n Model Predictive Controller Model Predictive Controller Desired trajectory for Difference from u(k) to converge to B reference trajectory CPS Week 2013 18

Stability Analysis q Stability: uClizaCon of all processors converge to set points q Derive stability condiCon à range of G q Tolerable variaCon of execuCon Cmes à Provides analyCcal assurance despite uncertainty CPS Week 2013 19

Stable System 1 0.8 CPU utilization 0.6 0.4 0.2 0 0 50 100 150 200 250 300 Time (sampling period) P1 P2 Set Point execuCon Cme factor = 0.5 (actual execuCon Cmes = ½ of esCmates) CPS Week 2013 20

Unstable System 1 CPU utilization 0.8 0.6 0.4 0.2 0 0 100 200 300 Time (sampling period) P1 P2 Set Point execuCon Cme factor = 7 (actual execuCon Cmes = 7 Cmes esCmates) CPS Week 2013 21

Stability q Stability condiCon à tolerable range of execuCon Cmes AnalyCcal assurance on uClizaCons despite uncertainty Predicted bound for stability actual execution time / estimation CPS Week 2013 22

FC-ORB Middleware Model u ( k ) ⎡ ⎤ 1 Predic?ve Measured ⎢ ⎥ r ( k ) Control u ( k ) ⎡ ⎤ Controller Output 1 2 ⎢ ⎥ Input ⎢ ⎥ r ( k ) u ( k ) ⎢ ⎥ ⎣ ⎦ 2 ⎣ ⎦ 3 Feedback lane Rate Rate Rate Modulator Modulator Modulator Priority Priority Priority Manager Manager Manager U?liza?on U?liza?on U?liza?on Monitor Monitor Monitor Remote request lanes Remote request lanes X. Wang, C. Lu and X. Koutsoukos, Enhancing the Robustness of Distributed Real-Time Middleware via End-to-End Utilization Control, IEEE Real-Time Systems Symposium (RTSS'05), December 2005. CPS Week 2013 23

Workload Uncertainty disturbance from periodic tasks Cme-varying execuCon Cmes CPS Week 2013 24

Processor Failure 1. Norbert fails. 2. move its tasks to other processors. 3. reconfigure controller 4. control u?liza?on by adjus?ng task rates CPS Week 2013 25

Summary: Model Predic?ve Control q ApplicaCon to CPU uClizaCon control q Robust uClizaCon control for distributed systems q Handle coupling among processors q Enforce constraints on task rates q Analyze tolerable range of execuCon Cmes q Applicable to many compuCng problems q MIMO: mulC-input (knobs), mulC-output (objecCves) q Coupling between objecCves q Constraints on knobs CPS Week 2013 26

Outline q CPU UClizaCon Control for Distributed Real-Time Systems q Model PredicCve Control q Thermal Control for Real-Time Systems q Nested Control Design CPS Week 2013 27

Nested Control q MulCple control objecCves q Coupling between objecCves q Dynamics at different Cme scales q Approach: Nested feedback control loop CPS Week 2013 28

Thermal Control for Real-Time Systems q Temperature control q Prevent processor overheaCng q Avoid hardware thro[ling à unpredictable slowdown q UClizaCon control q Maintain real-Cme performance q Enforce schedulable uClizaCon bound q UncertainCes q Power, ambient temperature, thermal faults, execuCon Cme 29 CPS Week 2013