Solving Fault Diagnosis Problems – a Computational Perspective

A. Varga

German Aerospace Center (DLR)

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Model-based fault detection and diagnosis (FDD)

u(t) y(t)

actuatorfaults d(t)w(t)

r(t)

residual signal

Decision makingResidual evaluator

Residual generator

θ(t) i(t)

evaluation signal

decision signal

FDD System

SensorsActuators Plantua (t) ym (t)

sensorfaults

parametricfaults

sensor signals

control signals

plant inputs

plant outputs

Open-loop System

noise inputs

disturbance inputs

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Numerical computations in model-based FDD

Development of synthesis models: trimming, linearization, model reduction, conversions, identification, …

Synthesis of linear residual generators (or fault detection filters): observer synthesis (pole assignment, H2/∞

- filtering), μ-synthesis, …

Integrated tuning of the FDD-system: multi-objective optimization based tuning of FDD system parameters to minimize false alarm rate, missed detection rate, detection times, …

Global assessment of the FDD-system: robustness assessment using Monte-Carlo simulations, global optimization based worst-case search, μ-analysis, …

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Numerical awareness in fault detection

Surprisingly … there are very few generally applicable methods !

Evidently … all tolerance-based numerical computations can go wrong !

Unfortunately … numerical algorithms can go wrong without notice !

Perfidiously … bad algorithms can sometimes produce good results !

Sadly … most algorithms in fault detection are bad !

How could we improve numerical awareness in fault detection ?

Special Issue Feb. 2004

Special Issue Feb. 2004

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Outline of talk

I. Nullspace-based synthesis approaches

II. Basics of numerical algorithms

III. An integrated synthesis algorithm

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Outline of talk

I. Nullspace-based synthesis approaches– exact/approximate synthesis problems – nullspace based simplifications– least order synthesis– synthesis procedures

II. Basics of numerical algorithms

III. An integrated synthesis algorithm

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Synthesis problems

1. Exact synthesis of fault detection filters (FD): nullspace method, parity space method, unknown input observer methods, geometric synthesis methods, …

2. Exact synthesis of fault detection and isolation filters (FDI): structured residuals (e.g., via a bank of detectors), directional residuals via model-matching, …

3. Approximate synthesis of fault detection filters (AFD): H- /H2/1

- optimal synthesis

4. Approximate synthesis of fault detection and isolation filters (AFDI): H2/1

- optimal model-matching synthesis

For all synthesis problems the nullspace method will play a central role in simplification of the solution process!

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Input-output system model with additive faults

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Residual generator

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Exact synthesis problems: no noise inputs

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Approximate synthesis problems: with noise inputs

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Interpretation of disturbance inputs

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NullspaceNullspace based synthesis based synthesis

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Simplified problemSimplified problem

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Simplified synthesis problems

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DiscussionDiscussion

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Polynomial vs. rational basesPolynomial vs. rational bases

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FD: Least order synthesisFD: Least order synthesis

Numerically reliable algorithm: V (2003)

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FD: Least order synthesis with scalar residuals FD: Least order synthesis with scalar residuals

Numerically reliable algorithms: V (2007)

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FDI: Least order synthesis with structured residualsFDI: Least order synthesis with structured residuals

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FDI: Least order synthesis with structured residualsFDI: Least order synthesis with structured residuals

Numerically reliable algorithms: V (2007)

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FDI: Least order synthesis with directional residualsFDI: Least order synthesis with directional residuals

Numerically reliable algorithms: V (2004)

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AFD: Optimal synthesisAFD: Optimal synthesis

Numerically reliable algorithms: V (2009), Glover & V (2011)

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AFDI: Optimal synthesisAFDI: Optimal synthesis

Numerically reliable algorithms: V (2005,2011)

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The The nullspacenullspace method historymethod history

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I. Nullspace-based synthesis approaches

II. Basics of numerical algorithms– computing with finite precision– satisfactory algorithms– conditioning of model representations

III. An integrated synthesis algorithm

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Finite precision computation

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Effects of roundoff errors

forward error

aa

a+a+δδ

DD

f*(a)

f(a)

f f ((DD))backward error

Algorithm: a ∈ D ⇒ f(a) ∈ f(D)Implementation: a ∈ D ⇒ f∗(a)

ff*

f

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Conditioning

Backward and forward errors for a backward stable algorithm f : R → R

Example: Multiple eigenvalues are extremely ill-conditioned!

The conditioning of a problem measures the sensitivity of the solution to perturbations in the problem data and is independent of any computational algorithm.

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Example: Caveats of finite precision computationsEigenvalue-based stability check for a matrix A:Re λ(A) < 0 or Re λ(AT) < 0

MATLAB example: (Nick Trefethen)A = -gallery('grcar',150);plot(eig(A),'k.')plot(eig(A'),'ro')

Causes of accuracy loss:- Computation with finite precision- Ill-conditioned eigenvalues

Finite precision computations on sensitive problems often can not be performed accurately!

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Numerical stability, conditioning & accuracy

The most we can ask for:Numerically backward stable algorithm applied to a well-conditioned problem produces guaranteed accurate results.

We have to live with:Numerically backward stable algorithm applied to an ill-conditioned problem can produce inaccurate results.

Be aware of:Numerically unstable algorithm applied to a well-conditioned problem can produce inaccurate results.

Less known fact:Unstable methods can be reliable if the instability can be detected!

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Recipe to develop a bad algorithm! (not to be followed)

Use intrinsically sensitive model representations (polynomial, rational)

Perform unstable computations (ill-conditioned transformations to canonical forms, raising matrices to powers, forming matrix products, inversions, …)

Use special tricks:

subtract similar size quantities contaminated by errors (catastrophic cancellation)

produce small final results using large intermediate quantities

ignore problem structure

ignore potential under- and overflows

To develop good algorithms, the negation of all above issues is necessary but … not sufficient!

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Input-output vs. state-space models

Not suited for numerical computations due to high intrinsic sensitivity of polynomials based system representations!

Better suited for numerical computations and must be preferred for algorithm development!

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“The perfidious polynomial” (Wilkinson)

g(s) =1

s25 + 325s24 + . . .+ 25!(Kressner)

% coefficients-based realizationpol = -25:-1;g = tf(1,poly(pol));plot(eig(g),'ro')plot(pol,0*pol,'k.')

Causes of accuracy loss:- Finite precision representation- Ill-conditioned polynomial roots

Model representations relying on polynomials are intrinsically very sensitive, thus not appropriate for reliable numerical computations

g(s) =1

(s+ 1)(s+ 2) · · · (s+ 25)

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Steer-by-Wire Control System (Odenthal & Bünte)

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Modeling of a steer-by-wire control system using polynomial/rational function manipulation

33.33 s^27 + 8.46e5 s^26 + 1.009e10 s^25 + 7.519e13 s^24 + 3.927e17 s^23 + 1.529e21 s^22 + 4.607e24 s^21 + 1.101e28 s^20 + 2.124e31 s^19 + 3.345e34 s^18 + 4.336e37 s^17 + 4.651e40 s^16 + 4.141e43 s^15 + 3.063e46 s^14 + 1.88e49 s^13 + 9.547e51 s^12 + 3.987e54 s^11 + 1.358e57 s^10 + 3.73e59 s^9 + 8.132e61 s^8 + 1.379e64 s^7 + 1.779e66 s^6 + 1.71e68 s^5 + 1.226e70 s^4 + 6.822e71 s^3 + 2.952e73 s^2 + 8.313e74 s + 2.061e76

------------------------------------------------------------------------------ 1 s^29 + 2.539e4 s^28 + 3.029e8 s^27 + 2.258e12 s^26 + 1.18e016 s^25

+ 4.597e19 s^24 + 1.386e23 s^23 + 3.316e26 s^22 + 6.402e29 s^21 + 1.01e33 s^20 + 1.311e36 s^19 + 1.409e39 s^18 + 1.259e42 s^17 + 9.345e44 s^16 + 5.766e47 s^15 + 2.948e50 s^14 + 1.243e53 s^13 + 4.294e55 s^12 + 1.204e58 s^11 + 2.706e60 s^10 + 4.819e62 s^9 + 6.716e64 s^8 + 7.289e66 s^7 + 6.212e68 s^6 + 4.194e70 s^5 + 2.152e72 s^4 + 7.713e73 s^3 + 2.094e75 s^2 + 1.008e76 s

G11 = -((G1*(S1-G2*G4*S1-C22*G3*G4*S1+C13*S1*S2*S4...-C13*G2*G4*S1*S2*S4-C13*C22*G3*G4*S1*S2*S4...+C12*C23*G3*G4*S1*S2*S4+S3*S4-G2*G4*S3*S4-C22*G3*G4*S3*S4)).../(-S1+G2*G4*S1+C22*G3*G4*S1+C11*G1*S1*S2*S4-C11*G1*G2*G4*S1*S2*S4...+C12*C21*G1*G3*G4*S1*S2*S4-C11*C22*G1*G3*G4*S1*S2*S4-S3*S4...+G2*G4*S3*S4+C22*G3*G4*S3*S4-G1*S1*S3*S4+G1*G2*G4*S1*S3*S4...+C22*G1*G3*G4*S1*S3*S4));

Extremely large range of coefficient values ⇒ severe numerical difficulties to be expected in all subsequent computations

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Conditioning of state-space models

Which characteristics define model conditioning?– numerical range of data – conditioning of eigenvalues/eigenvectors of A– controllability/observability balancedness– nearness to non-minimality

How model conditioning can be improved ?– using similarity transformations (scaling, balancing)– model order reduction

Improper scaling can cause the failure of any computation!

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What is a satisfactory algorithm ?Generality: Applicability without restrictions

Reliability: Numerical stability

Efficiency: O(n3) floating point operations (flops)

Other adjectives: simple, beautiful, elegant, robust, fast, accurate, polynomial, exponential, embedded, compact, top10, Holly Grail, honest, “comme il faut” … some with negation

A satisfactory numerical algorithm can serve as basis for a general purpose routine which meets the standards of quality numerical linear algebra or control software (e.g., LAPACK or SLICOT).

A satisfactory synthesis algorithm for solving fault detection problems must exclusively rely at each computational step on satisfactory numerical algorithms and structure exploiting computations.

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I. Nullspace-based synthesis approaches

II. Basics of numerical algorithms

III. An integrated synthesis algorithm– integrated algorithms – nullspace method in state-space formulation – comparison of methods

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Integrated synthesis algorithms

Typical algorithmic approaches in control and fault detection:- composition of modularized computational steps- problem structure between steps not exploited

Integrated approach: computational steps connected at a finer granularity level by exploiting all structural information at the termination of each step

Expected advantages:- more efficient computations (less overhead, minimal storage)- increased reliability (structural aspects fully exploited)

Caveats: more involved algorithmic development; careful analysis of structural features needed; restricted modularization/reusability

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Example: Integrated algorithm for FD filter synthesis

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Choice of synthesis model

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S1. Computation of proper rational S1. Computation of proper rational nullspacenullspace basesbases

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S1. Computation of proper rational S1. Computation of proper rational nullspacenullspace basesbases

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S1. Computation of proper rational S1. Computation of proper rational nullspacenullspace basesbases

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S2. LeastS2. Least--order synthesisorder synthesis

Numerically reliable algorithms: V (2003,2004)

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S3. StabilizationS3. Stabilization

Numerically reliable algorithms: V (1995,1998)

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Satisfactory synthesis algorithm

Generality: A solution can be computed whenever one exists

Reliability: Relies on numerically stable or numerically reliable algorithms

Efficiency: O(n3) floating point operations

The nullspace method for the synthesis of fault detection filters is a satisfactory synthesis algorithm.

What we can say about other methods?

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Parity space method

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Parity space method

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Observer-based synthesis

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Observer-based synthesis

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Comparison of methods (V, 2009)

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Concluding remarks

A new generation of numerically satisfactory integrated algorithms based on detector updating techniques has been developed to solve the main classes of FD/FDI synthesis problems.

The nullspace method has a central role in simplifying the synthesis problems and in addressing the least order synthesis aspect.

Robust numerical software to perform all basic computations is available in the MATLAB/SLICOT-based Descriptor Systems Toolbox (V, 2000-2011) (licensed by SYNOPTIO)

All synthesis algorithms are implemented in the Fault Detection Toolbox (V, 2006-2011) (proprietary software of DLR)