2001 MURI Mathematics of Failures in Complex Systems

1. 2001 MURI Mathematics of Failures in Complex Systems Project Title: Characterization and Mitigation of Service Failures in Complex Dynamical systems Technical Vision and Approach Program manager:Dr. Robert Launer (launer@arl.aro.army.mil) Mathematical and Computer Sciences Division U.S. Army Research Office, P.O. Box 12211 Research Triangle Park, NC 27709-2211 Principal Investigator:Professor Asok Ray (axr2@psu.edu) The Pennsylvania State University University Park, PA 16802

2. Predict • Avoid • Adjust • Reorganize • Fix Complex System Failures Software Hardware Networks Platforms Understanding Failure Achieving Success

3. AWACs UAV R (S1) 1 SA-6 SA-6 T 1 C2 SA-6 Airport R 2 T 2 KC-10 C2 SA-6 (F1) NEW Factory SAM SEAD R SA-6 3 Factory (J1) T 3 Airport SA-12 Train station RIVETJOINT Factory (S2) MITIGATION OF PERVASIVE FAILURES Man & Machine Command & Control of Battlefield Dynamics Ref:DARPA Information Technology Office

4. Pervasive Fault Tolerance of Hierarchically Structured Human-Engineered Systems Failure Simulation Network Collaboratory PROJECT GOALS • Failurecharacterization • Continuous and discrete hardware faults • Software faults • FailureMitigation via active and passive control • On-line and off-line system reconfiguration • Gracefully degraded operation • Experimental validation of theoretical results with hardware in the loop • Collaborative research and training of participants • from academia, government, and industry • Failure Data and Information Repository

5. MODELING AND CONTROL OF PERVASIVE FAILURES • FailureCharacterization • Physics-based dynamic modeling of continuous faults - Damage in mechanical structures • Semi-empirical Modeling of hard failures and soft faults - Malfunction of electromechanical and electronic hardware -Malfunction of communication and control software - Human-machine operation faults • Integration of physics-based and semi-empirical models • Failure Mitigation • Continuously-varying robust estimation & control • Discrete-event robust decision & control • Hierarchically structured hybrid decision & control

6. OBJECTIVES OF: Pervasive Failure Modeling • Localization of Potential Failure Source(s): benign and malignantfaults • Detection andIdentification of Incipient Failures: malignant faults • Failure andDamage Prediction under Anticipated Operation: prognosis • Failure-Accommodating Robust Decision & Control: graceful degradation

7. PHYSICS-BASED MODELING OF FAILURES • Nonlinear Stochastic Dynamics of (Inhomogeneous) Complex Processes • Multi-Scale Nonstationary Features of Temporal and Spatial Parameters • Non-Colocated Sensory Information • Real-time Information Filtering

8. SEMI-EMPIRICAL MODELING OF FAILURES • Computer Systems Software and Hardware • Performability and survivability analysis • Software aging and rejuvenation • Discrete- and continuous-state representation • Electromechanical and Electronic Hardware • Fault Manifestation Analysis • Statistical Failure Analysis

9. TECHNICAL CHALLENGES: INTEGRATION OF PHYSICS-BASED AND SEMI-EMPIRICAL FAILURE MODELS • Nonstationary Statistics of Discrete Events Exciting Nonlinear Dynamics • Complexity of Stochastic Analysis via Monte Carlo Simulation • Robustness ofMulti-Scale Nonstationary Distributed Decision & Control Systems • Real-time Information & Control Systems

10. TECHNIQUES OF APPLIED MATHEMATICS • Systems Sciences: Functional Analysis • Nonlinear time-varying dynamical systems • Fractal geometry and fractional-dimensional processes • Wavelet decomposition of nonstationary random signals • Stability analysis and decision & control synthesis • Resource-bounded optimization • Markov and semi-Markov failure processes • Computer Sciences: Automata & Languages • Finite-state automata and regular languages • Discrete-event systems and hybrid control • Discrete and continuous (stochastic) Petri nets

11. TECHNICAL APPROACH: • Multi-Scale Nonstationary Modeling • Identification andQuantification of Failure Behavior • InformationFusion ofNon-Colocated Sensor Data and Faulty Process Model

12. Fatigue Cracks in Tube Walls Creep Thinning in Tube Walls TYPICAL DAMAGE IN MECHANICAL STRUCTURES

13. 24 Thickness 3.175 Frequency 22 of Loading 25.4 10 hz 20 18 Material: 7075-T6 alloy Crack Length (mm) 16 14 320.67 12 60 Specimens S = 70.65 MPa 14.288 10 max R = 0.6 8 160.33 0 2 4 6 8 10 12 14 9.525 DIA 4 Number of Cycles x 10 6 Holes Test P (kN) R max 1 22.79 0.6 25.4 2 22.25 0.5 All Dimensions in mm 3 15.19 0.4 Three sets of 60 carefully controlled tests on specimens made of 7075-T6 alloy RANDOM FATIGUE TEST DATA Ghonem and Dore (1987) .

14. DYNAMICS OF CHAOTIC MOTION Forced van der Pol Equation Steady-state numerical solution Approximately 22.5 response cycles Subharmonic Number  50/22.5  2.2 Five response cycles From t=0 to t=45.16

15. Dilated Self-similar Waveform (Deterministic) Statistically Scale-invariant Process (Identical Distribution) First Order Autoregressive Process (NOTStatistically Scale-invariant) Self-similarity and Scale-invariance Scaling Property: Self Similarity with Parameter H:

16. MULTIRESOLUTION WAVELET ANALYSIS Decomposition of Chirpy Noise Signal Using the db 4 Orthogonal Wavelet S=a1+d1=(a2+d2)+d1= = a6+d6+d5+d4+d3+d2+d1

17. MULTI-SCALE NONSTATIONARY DETERMINISTIC FAILURE MODELING • Failure Model Formulation and Order Reduction • System identification in time and/or frequency domain • Inverse modeling under ill-posed conditions • Recurrent neural network (with simulation data) • Singular perturbation for model order reduction • Nonlinear Time-Varying Dynamics of Fault Propagation • Chaotic behavior of the fault propagation process • Identification of fractal attractors and repellers • Sensitivity to the initial state of fault(s)

18. MULTI-SCALE NONSTATIONARY STOCHASTIC MODELING • Fractional Brownian Motion (fBm) • Random fractalrepresentation of fault characteristics • Potentially benign faults • Potentially malignant faults • Long-memory processes withself-similardisturbances • Nearly1/fsignals with fractional Gaussian noise (fGn) • Statistical wavelet analysis and synthesis • Statisticalself-similarityof propagated faults • Scale invariance • Wavelet shrinkage for fault characterization • Multivariate wavelet density estimation

19. IDENTIFICATION AND QUANTIFICATION OF FAILURE BEHAVIOR • Identification of Failure Events • Multi-level hypotheses testing • Pattern matching and scene analysis • Robust identification of uncertainty dynamics • Event generation for discrete-event modeling • Quantification of Damage Measure • Translation-variant s-finite deterministic measure • Hausdorff measure for fractional dimensional spaces • Continuous and discrete probabilistic measure

20. SYSTEM DEPENDABILITY • Performability • Reliability + Availability + Performance • Survivability • Reliability + Availability + Service • Safety • Security

21. Achieving SYSTEM DEPENDABILITY • Fault Forcasting • Fault Prevention • Fault Accommodation • Fault Removal

22. ANALYSIS OF SYSTEM DEPENDABILITY • Model-based Evaluation of System Dependability • Fault-tree analysis • Markov, Markov regenerative, andsemi-Markov analysis • Stochastic Petri net • Statistical inference • Self Similarity of Network Traffic • Modeling via fractional Brownian motion (fBm) • Multi-scale signal decomposition via wavelet transform

23. MITIGATION OF PERVASIVE FAILURES

24. MITIGATION OF PERVASIVE FAILURES • Passive Control of Software, Hardware, and Electronic and Electromechanical Components • Continuously-Varying Control of a Single Entity • Failure diagnosis and prognosis • Discrete-time robust output feedback control • Discrete-Event Decision & Control of Multiple Entities • • Robust and failure-accommodating decision & control • • Game-theoretic approach to systems engaged against others • Hybrid (i.e., continuous and discrete-event) Control • of Interacting Entities over Wide Ranges of Operation

25. Discrete-Event System (DES) Decision & Control Synthesis • Qualitative control of discrete event systems • Focusing on the order of event occurrence instead of the specific instant of their occurrence • Failure–accommodating controlled operation • Guaranteeing that the system meets the desired logical goals although operating in a (possibly) degraded mode

26. Plant Description Control Objectives Constraint: Plant FSM Model Go K Control Specifications SyncComp G||S Plant DFSM Model G Completion of S, i.e., S Is G||S Controllable? Y N Iteration: S’ S G||S’ controllable S is the Controller S’ is the Controller DISCRETE EVENT SUPERVISORY CONTROL SYNTHESIS

27. States Events q0idle and safe q6 aattack the target q1searching for target Aalarm q2alert (in danger) bpartly damaged q3engaged in combat t l q4partially damaged Cmission completed q5destroyed ddestroyed q0 q6back to the base Dsuccess/abort eescape A S/s landing to base l d C/e S/ssearch enemy/friend e A ttaking off from base A S/s q5 q1 q2 e e d d a b a d D b A b q4 q3 b a A SIMPLIFIED FINITE-STATE AUTOMATON MODEL OF ROTORCRAFT OPERATION

28. PERFORMANCE AND ROBUSTNESS OF CONTROLLABLE SUPERVISORS • Asigned real-valued measurepartitions an accepted language into positive, negative, and null sets • A distance function between two regular languages is defined based on the measure • A metric spaceof regular languages is constructed with the distance function • A design problem is to achieve a maximally performing controllable supervisor for the nominal plant model • A dual problem is to design a supervisor that is maximally robust, i.e., minimally sensitive to modeling uncertainties

29. highS1 highS2 High Level Controller Feature Selector #2 Feature Selector #1 highSc2 highSc1 Inverse Feature Selector #2 Inverse Feature Selector #1 . . Low Level Controller #2 Low Level Controller #1 lowSc2 lowSc1 lowS2 lowS1 Low Level Plant #2 Low Level Plant #1 MUTI-LEVEL HIERARCHICAL DECISION & CONTROL

30. UNIQUENESS OF THE HIERARCHICAL SUPERVISOR SYNTHESIS METHOD • Abstraction based on the behavior of the lower level closed-loop (controlled) system; • Extension of the controllability and language measure concept to multi-level hierarchical controller design; • Control specifications dependent on complexity of the plant model at the corresponding level of control hierarchy.

31. DAMAGE MITIGATING CONTROL OF COMPLEX SYSTEMS

32. DAMAGE MITIGATING CONTROL OF COMPLEX SYSTEMS • Motivation: To achieve high performance with increased: • Safety • Reliability • Availability • Maintainability • Objective: To ensure structural integrity by: • Reduction of material damage (e.g., fatigue cracking) • Simultaneous enhancement of performance via active control

33. INGREDIENTS OF REAL-TIME DAMAGE MITIGATING CONTROL • Damage Sensing Systems • Multiple damage sensors • ARMA model of damage propagation • Information fusion • Modeling uncertainty • Sensor noise • Hierarchical Decision & Control • Robust performance • Intelligent decision-making • Approximate reasoning for damage control • Discrete-event decision for operation & maintenance

34. DAMAGE MITIGATING CONTROL OF COMPLEX SYSTEMS • Technical Approach • To model the dynamics of structural degradation in: • Stochastic fractional-dimensional state-space • Discrete-event state space • To synthesize robust decision & control algorithms for: • Failure prognosis via statistical wavelets • Life extension via active control • Technology Transfer • To enhance the science & technology base of: • Rotorcraft and land-based vehicle industry • Gas turbine engine industry

35. Mission Management Level Vehicle Management Level Flight Control Level Wide-Range Nonlinear Damage Control Rotorcraft Structural Health and Usage Monitoring System . Robust Linear Parameter-Varying Output Feedback Control Analytical Measures of Damage States and State Derivatives Signal Conditioning and Signal Validation (FDIR and calibration ) Conventional and Special-Purpose Sensor Systems Actuator Dynamics Flight Dynamics and Structural Dynamics Information-Integrated Health Management and Damage Mitigating Control of Rotorcraft Note: Damage, leading to degradation or loss of vehicle safety, is represented by both continuous-varying and discrete-event states that include faults of electronic components and a variety of degradation in mechanical structures such as fatigue cracking, wear, spalling, and corrosion. However, damage measures are constructed to be C1-continuous, non-negative, finite, and monotonically increasing.

36. dam dam (k) str y (t) (k) y y Structural S u(t) Plant Model dyn y (t) Dynamics reg y (t) Damage H S S Prediction reg e (k) Model reg y (k) u(k) _ dyn y (k) Linear set y (k) _ . fb Gain- Scheduled u (k) + ref y D(k) dyn e (k) S + Controller S + . K(z) + S ff u (k) S Sample D(k) (k) H Hold Reference Fuzzy Signal Damage Generator Controller Linear parts of the control system RR(k) Nonlinear parts of the control system Wide- Range Fuzzy Damage-Mitigating Control

37. Life Extending Controller Fatigue Crack Damage Model Aeroelastic Model Damage Mitigating Control System Schematic Structural stresses Damage vector Control Input Actuator Model Stochastic State-space Model of Fatigue Crack Damage On-line Sensor Data Rigid-Body Model Damage Rate vector Aeroelastic Wing Model Propulsion Model Rigid-Body Aircraft PLA Pilot Commands Dynamic Model Atmospheric Model DAMAGE MITIGATING CONTROLOF A FIXED-WING TACTICAL AIRCRAFT Damage Prediction System

38. x b a y w x b s y ,y b s x w V z ,z s w z b TACTICAL AIRCRAFT SIMILAR TO F-15

39. 100 20 Reference Reference PC PC ) DMC1 50 15 DMC1 DMC2 DMC2 DMC2 0 10 Roll Rate (deg/sec) PC Pitch Rate (deg/sec DMC1 PC Reference Reference DMC1 - 50 5 DMC2 - 100 0 - 150 - 5 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 ) Time (sec ) Time (sec 4 1 PC Reference DMC1 3 PC DMC2 0.9 DMC1 ) 2 DMC2 0.8 DMC1 1 Reference PC 0.7 0 Sideslip Angle (deg 0.6 Crack Length (in mm) PC - 1 0.5 - 2 DMC1 0.4 DMC2 - 3 DMC2 0.3 - 4 0.2 - 5 0 2 4 6 8 10 12 14 0.1 0 5 10 15 20 25 30 35 Time (sec) Number of kilomaneuvers AIRCRAFT PERFORMANCE AND DAMAGE UNDER TURN REVERSAL MANEUVER

40. The Space Shuttle Main Engine (SSME)

41. SSME PROPULSION SCHEMATIC

42. 3200 3000 Pressure Range: 2100 psi to 3000 psi 2800 2600 Chamber Pressure (psi) 2400 2200 2000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 6.06 6.04 Pressure Range: 2100 psi to 3000 psi -3 x 10 Mixture Ratio 2.5 6.02 Reference With Damage Control 2 With Damage Control Without Damage Control 2 6.00 Pressure Range: 2100 psi to 3000 psi /H With Damage Control Without Damage Control 2 1.5 O 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Reference Without Damage Control 5.98 Damage in Turbine Blades 1.0 With Damage Control -5 x 10 2.5 Without Damage Control 0.5 Fuel(H2) Turbine 2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.5 Time (sec) Damage in Turbine Blade 1 0.5 Pressure Range: 2100 psi to 3000 psi 0 0 0.2 0.4 0.6 0.8 1.0 1.2 Oxidant (O2) Turbine

43. VALIDATION OF NEW DMC CONCEPTS IN LABORATORY ENVIRONMENT • Failure Simulation Laboratory • Fatigue Testing Apparatus • Aircraft Simulation Testbed • Rocket Engine Simulation Testbed • Fossil Power Plant Simulation Testbed • Rotorcraft Excellence Center • Rotorcraft Simulation Testbed • Aeroelasticity Simulation Testbed • Health and Usage Monitoring (HUMS) Testbed • Computational Fluid Dynamics Laboratory • Combustion Simulation Testbed • Gas Turbine Engine Simulation Testbed • Rocket Engine Simulation Testbed

44. Break

45. COMPLEX SYSTEM FAILURES Software Hardware Networks Machinery Achieving Success Understanding Failure • Predict • Fix • Avoid • Adjust • Reorganize

46. MATHEMATICAL MODELING OF FAULT GENERATION AND PROPAGATION • Fault Propagation Models • Physics-based modeling • Semi-empirical modeling • Measures of Pervasive Fault Tolerance • Physics-based measures • Information-theoretic measures • Hierarchically Supervised Automata • Hybrid decision & control for failure mitigation • Quantitative evaluation of robust performance

47. PLANT DYNAMICS, CONTROLLER, AND INFORMATION GENERATOR Hierarchical Discrete Event Controller SupervisoryController Information Generator Hierarchical Controller Other Weapon System Controller Aircraft Controller Discrete Event Routing Clustering Routing Clustering Filtered Information Events Control Decision Support Interface Controller Interface Controller Interface Decision Support Interface Event Generator Information Filter Action Generator Action Plant State Event Generator Generator Filter Dispatcher (Plant Interface with Control) Dispatcher (Simulation Interface and Control) Plant Information Plant Control Plant Information (Simulated) Plant Dynamics Plant (Simulator) Platform Simulation Platform Simulation Platform Simulation Platform Simulation Platform Simulation Platform Simulation An example of System Complexity: INTELLIGENT BATTLEFIELD AUTOMATION NOISE/UNCERTAINTY ACCOMMODATION Sensor information validation and calibration Noise modeling at the interface level Noise masking for event/action Generators PLANT/CONTROLLER INTERFACE Event/action generators serving as continuous/discrete interfaces Accommodation of multiple controllers with various plant subsystems HIERARCHICAL AGGREGATION Feature selector for generating meta- language for the supervisory Controller Inverse feature selector for control actions CONTROL SYNTHESIS AUTOMATION Assuring controllability, observability, scalability, and hierarchical consistency  JAVA-based controller synthesis tools

48. DAMAGE MITIGATING CONTROL • High performance with increased: • Fault tolerance • Damage tolerance • Enhanced reliability and safety via: • Reduced structural damage • Information-based maintenance • Synergistic combination of: • Systems Science • Computer Science • Mechanical Science • Material Science