experimental certification of jet engine...

ETH, April 3, 2008

Experimental Certification of Jet Engine Controllers: Generalized Stability Margin Inference for a Large Number of MIMO Controllers

Professor Bob BitmeadDepartment of Mechanical & Aerospace EngineeringUniversity of California, San Diego

Acknowledgments to: Jisang Park, Abbas Emami-Naeini, Sarah McCabe, Dan Urban, Gonzalo ReyDARPA SBIR Program, Pratt & Whitney, SC Solutions, Impact Technologies

Tuesday, April 3, 2012

ETH, April 3, 2008

Motivation The demand for efficiency and

performance pushes engine operation to its physical limits

(Many constraints to meet to avoid rapid degradation and operational instabilities)

Joint Strike Fighter(Short Take Off/Vertical Landing)

Temp. limit for turbine life

The maximum efficiency for a compressor andfan is attained at near stall and surge margin.(may cause engine operation instability)


ETH, April 3, 2008

Outline and Background

I. Controller Certification Single uncertain MIMO engine with a fixed model A large number of possible constraints

must be satisfied, but a good design method 20 constraints and up to 4 of constraints are active at any

one instance, 6196 possible combinationsII. Less Conservative Margin and -gap ComputationsIII. Experiments for Estimation of the Margin with an error bound


ETH, April 3, 2008

Jisang Park, Robert R. Bitmead, “Controller Certification”, Automatica, vol. 44 no. 1, pp. 167 ~ 176, Jan. 2008.

Controller Certification

Looking for the best tasting wines by sampling ...

I cannot test all the wines here

although I would like to

Which wines should I test?

ideas of closeness - metric distance

informative choices for testing

quality measures and inference for neighbors


bP,C ! [0, 1]

ETH, April 3, 2008

The Main Tools Generalized Stability Margin

Large margin means very robust, smaller margin less robust Connection to performance

bP,C

T (P, C) =

!

P (I + CP )!1C P (I + CP )!1

(I + CP )!1C (I + CP )!1

"

bP,C =

!

(!T (P, C)!!

)"1 if T (P, C) is stable,0 else.

bP,C,! =

!

(!max [T (P, C, ")])!1 if T (P, C) is stable,0 else.


ETH, April 3, 2008

The Main Tools

Generalized Stability Margin Genuinely MIMO Extends SISO single-loop measures from legacy systems

Not really true and still needs validation For us,

and may be computed from experimental data using frequency-domain system identification

Requires the estimation of a transfer function infinity-norm or maximal singular value

Even for SISO systems, is a MIMO calculation

bP,C = 0.3 ! Gain Margin=2.7dB, Phase Margin=15 deg

bP,C bP,C,!

bP,C


ETH, April 3, 2008

The Main Tools

Vinnicombe ν-gap metric

A true metric measuring the distance between MIMO transfer functions A scaled variant on the distance between Nyquist plots in SISO case

A similar condition involving P and C arises in MIMO Nyquist stability Frequency-by-frequency variant

!!(C1, C2) =

!"

"

"(I + C2C

"

2 )!1

2 (C1 ! C2)(I + C"

1C1)!1

2

"

"

"

!

, if WNC holds

1, else

WNC:

!

det (I + C!

2C1) (j!) != 0,"!, andwno det(I + C!

2C1) + "(C1) # "(C2) = 0

!!(C1, C2, ") =

!

#max"

(I + C2C"

2 )!1

2 (C1 ! C2)(I + C"

1C1)!1

2

#

(j"), if WNC holds

1, else


ETH, April 3, 2008

The Main Tools

!!(C1, C2) ! [0, 1]

Vinnicombe ν-gap metric Easily computable via matlab “gapmetric.m”

Large ν-gap means large distance Inherently MIMO quantity Extends the Gap Metric into an easily computable value Familiar

Almost intuitive


ETH, April 3, 2008

Main Inferences Vinnicombe’s stability and margin guarantees

is stable, if is stable and

Frequency-by-frequency version

is experimentally measurable is easily computable can be inferred

Stability / Margin Guarantees

(P, C1)

!!(C1, C2) ! [0, 1]bP,C ! [0, 1]

arcsin bP,C2,! ! arcsin bP,C1,! " arcsin !"(C1, C2, ")

bP,C1,!

bP,C2,!

!!(C1, C2, ")


ETH, April 3, 2008

Controller Certification

A controller is said to be certified if its generalized stability margin is guaranteed at a sufficient level in operation with the plant

The -gap metric and bP,C are the central tools inherently able to handle MIMO system frequency-by-frequency test easy computability – “gapmetric.m”

Starting Point Single uncertain engine, , with a fixed model, A large set, , of candidate controllers, Every is designed to achieve internal stability with Selection of a small subset, , so that through

experimental testing of pairs , the complete set is certified.


ETH, April 3, 2008

Mathematical Definitions

A controller is said to be certified at level if, using experimental data with the unknown actual plant , we can guarantee

A controller is said to be rejected at level , if we can guarantee


ETH, April 3, 2008

Main Inferences Vinnicombe’s stability and margin guarantees

is stable, if is stable and

Refined Inferences We can infer from and

This requires no additional test on

We can guarantee from , without extra test, if

Stability / Margin Guarantees

(P, C1)

arcsin !!(C1, C2) < arcsin bP,C2! arcsin"

bP,C1> !

(P, C2)

bP,C2! ! bP,C1

! !


arcsin !!(Ci, Cj) ! arcsin" " arcsin bP,Ci

ETH, April 3, 2008

Certification Algorithm (Finite Controller Set)

Step 1 (Search) For each uncertified controller, , with count the number of uncertified controllers, , that satisfy

Then choose the controller, , with most controllers, , satisfying this

Step 2 (Experiment) Do the experiment on to retrieve from closed-loop data

Step 3A (Certifying) If , certify all controllers, , satisfying

Step 3B (Reject) If , reject the controllers, , satisfying

Iterate from Step 1 to Step 3 until all controllers are certified or rejected.

arcsin !!(Ci, Cj) < arcsin bP̂ ,Ci! arcsin "

arcsin !!(Ci, Cj) < arcsin bP,Ci! arcsin"


ETH, April 3, 2008

Numerical Examples

Stabilizing Region of the Actual Plant, . (Unknown)

Stabilizing Region of the Nominal Model, .

Parametrized controllers,

Stabilizing regions for , and 74 selected controller parameters


ETH, April 3, 2008

Numerical Examples

Stabilizing Region of the Actual Plant, . (Unknown)

Stabilizing Region of the Nominal Model, .

Parametrized controllers,

Stabilizing regions for , and 74 selected controller parameters

Not a stabilizing region for the actual plant


ETH, April 3, 2008

Numerical Examples (stability certification)

1. (Search) For each uncertified controller, , count the number of uncertified controllers, , satisfying

(1)

If we choose as a test controller, then we could expect there are 34

controllers satisfying

which means for those controllers

Numbers of controllers satisfying (1) after first searching

arcsin !!(Ci, Cj) < arcsin bP,Ci

arcsin !!(C37, Cj) < arcsin bP̂ ,C37


ETH, April 3, 2008


2. (Experiment) Do the experiment on the plant-controller pair to retrieve from closed-loop data.

3. (Certifying) Actually 21 controllers satisfy

21 certified controllers by the first experiment

arcsin !!(C37, Cj) < arcsin bP,C37


ETH, April 3, 2008


Iteration for stability certification After 31 tests, stability certification for 60 controllers (blue circles) was

completed.

arcsin !!(Ci, Cj) < arcsin bP,Ci


ETH, April 3, 2008

Numerical Examples (margin certification)

27 controllers certified as , by 7 experiments.


ETH, April 3, 2008

Conservatism in Computations

The generalized stability margin and the -gap metric computations involve the maximum singular value over all frequency points A supremum over frequency A scalar measure of a matrix property


ETH, April 3, 2008

Simultaneous Scalings

For an efficient controller certification, larger and smaller should be incorporated if possible


ETH, April 3, 2008

Problem Setting

, positive definite symmetric matrices, constant over frequency

, positive definite hermitian matrices at a fixed frequency

! ! !


ETH, April 3, 2008

Problem Setting

Scaling for computation: convex optimization Scaling for - gap computation: non-convex optimization Simultaneous scaling selections for and - gap (XY-Centering algorithm, Skelton and Iwasaki)

Stability is invariant to scalings

Scalings have a stability analysis purpose (In SISO systems, GM/PM is invariant to scalings)


ETH, April 3, 2008

F100 Jet Engine Certification – Use of Scalings

Constant diagonal input and output scalings (Wi, Wo) for less conservative margin computations

The F100 engine model was chosen at sea level static flight conditions with power lever angle (PLA) of 36 degree

Reduced order model is used in controller design The model-stabilizing multivariable Proportional-plus-Integral controller

is designed in the LQR framework

where, Kp and KI are 5x5 gain matrices The first certification parameter is Kp(1,1) and the second one is KI(2,2) Each parameter changes up to from its original value

arcsin !!(W!1

i CiW!1

o , W!1

i CjW!1

o ) < arcsin bWoP̂Wi,W!1

iCiW

!1

o! arcsin"

±30%


ETH, April 3, 2008

F100 Jet Engine Certification – Use of Scalings

a), b): 8 tests complete certification of 81 controllers for c), d): all controllers are rejected at level of


ETH, April 3, 2008

Implementation of Controller Certification

Development of GUI in corporation with SC Solutions


ETH, April 3, 2008

I. Controller CertificationII. Estimation of the Generalized Stability

Margin


ETH, April 3, 2008

Introduction to the bP,C estimation

The stability margin inference of relies on the estimated value of

The generalized stability margin will be estimated from the empirical transfer function estimate (ETFE) of T(P,C)

A scalar standard test signal for the error bound analysis – a probing input signal over a specific range of frequency,

The same scalar test signal used in the experiment design for the MIMO ETFE


ETH, April 3, 2008

Estimating Frequency Response

Estimating one frequency response value - SISO case Input / Output

Model

DFTs

Empirical Transfer Function Estimate

SISO Error Analysis


ETH, April 3, 2008

SISO Error Analysis

Standard Test Signal,

a priori information Inputs:

Impulse response: Noise power:

Pre-experiment (Nr) to reduce the initial conditions (De Vries, 1994) Chirp up & down (N) - manageable probing frequency range (wide then narrow) - smooth (not multi-sine) -

y(t) recording starts

Probing from DC to 5 Hz


ETH, April 3, 2008

SISO Error Analysis

Dependencies Number of data Ringdown time of engine , impulse response bound Number of “pre-experiment” of the standard test excitation Input energy in band Noise energy in band Bounds on pre-input and test inputs


ETH, April 3, 2008

SISO to MIMO

If input signals, then need to perform separate experiments Each has its own pre-experiment Define an matrix Each has the vector input signal

Same scalar standard test signal Matrix of all experiments has full rank Empirical Transfer Function Estimate(MIMO)

where are the vector DFTs of output vectors, , and input vectors, , respectively. MIMO error bound


ETH, April 3, 2008

Single Frequency Result to Neighboring Frequencies Bound

In order to bound the error between two fixed frequency points Interpolation error and dependencies

with and


ETH, April 3, 2008

Estimating bP,C

Conduct closed-loop identification experiment for MIMO transfer function

Scale at each frequency using earlier results(optional for certification)

Take max value plus bound


ETH, April 3, 2008

Numerical Example (Experiment Design)

Linearized MAPSS Model: a transfer function matrix Using Hadamard matrix, , and a standard test input, .

6 consecutive experiments with the standard test inputs

6 consecutive standard test inputs for the first channel


ETH, April 3, 2008

Numerical Example

Matrix DFT:

The maximum singular value plot of a) and b)ETFE of

a)

b)


ETH, April 3, 2008

Numerical Example

Effect of different length of pre-experiments,


ETH, April 3, 2008

Noise Reduction in ETFE

Downsampling the ETFE Take N=MxP samples of data, i=0,1,...,MP-1 Compute the N-point DFTs and ETFE Extract every Mth point, i=0, M-1, 2M-1,..., MP-1 from the ETFE The effective noise power at these M frequencies is reduced by a

factor P ETFE estimation error is improved at each frequency interpolation error is increased because we have ETFE

samples only at M points Tradeoff


ETH, April 3, 2008

Conclusions

Development of the stability inference tool Identifying the -gap metric as a MIMO controller certification tool Providing certification/rejection algorithms

LMI formulations for the generalized stability margin and the -gap metric computations for efficient MIMO controller certification Reducing conservatism in computations of margin and metric

individually and jointly through scalings

Experiment design for estimation of the generalized stability margin with an guaranteed error bound Derivation of the error bound Designing inputs for MIMO ETFE (pre-experiments)


ETH, April 3, 2008

Further Work

Certification algorithm of an infinite set of MIMO controllers

Certification for fleet variability

Time variation in plants and/or controllers

A direct method to certify a non-linear plant and controller pair

Multi-objective optimization for maximizing the difference between the scaled generalized margin and -gap metric

Methodology for retrieving the upper bound on impulse response empirically


ETH, April 3, 2008

Implementation of Controller Certification

Development of GUI in corporation with SC Solutions

I. Controller Certification II. Less Conservative Margin and Vinnicombe-gap Computations III. Experiments for Estimation of the Margin


ETH, April 3, 2008

Certification for JSF engine

DoD SBIR

F135 test stand

Test data

Acknowledgment


ETH, April 3, 2008


DoD SBIR

F135 test stand $Test data

Acknowledgment


ETH, April 3, 2008


DoD SBIR

F135 test stand $

$

Test data

Acknowledgment


ETH, April 3, 2008


DoD SBIR

F135 test stand $

$¢

Test data

Acknowledgment


ETH, April 3, 2008

Simultaneous Scaling of and -gapTheorem 3 At a fixed frequency , consider the frequency response of a plant model

stabilizing controllers and and consider positive definite hermitian matrices and with and . If a solution exists with achieved objective value and for the LMIs,

and

with following properties,

or

Then the scaled at is bounded below by and the scaled -gap metric at is bounded above by

and

! ! !


ETH, April 3, 2008

Numerical Example

Scalings for maximizing

Simultaneous scalings increased the generalized stability margin and decreased the -gap


ETH, April 3, 2008

Frequency Dependent Scalings of bP,C

Theorem 2.1 At a fixed frequency , consider positive definite hermitian matrices and with and . If a solution of the

following optimization, minimize

subject to

exists with achieved objective value , then the scaled at is bounded below by ,

Yi, Xo


ETH, April 3, 2008

Constant Diagonal Scalings of bP,C

Corollary 1. Consider constant positive definite hermitian matrices and

with and let be a state space realization

of . If a solution of LMI optimization,

minimize

subject to

exists with achieved objective value , then the scaled is bounded below by

over all frequencies

M


ETH, April 3, 2008

Scaled -gap

Using and , minimization of the scaled -gap can be formulated as

minimize

subject to

Scaled -gap metric


ETH, April 3, 2008

LMI Formulation for Scaling -gap Using Schur complement, Minimize

subject to

Minimize

subject to


ETH, April 3, 2008

LMI Formulation for Scaling -gap Using Schur complement, Minimize

subject to

Minimize

subject to

Iterative minimization on and


ETH, April 3, 2008

Define the following matrices and ,

then define two sets and ,

LMI Formulation for Scaling -gap


ETH, April 3, 2008

LMI Formulation for Scaling -gap

Theorem 2 At a fixed frequency , consider the frequency response of a plant model , stabilizing controllers and and consider positive definite hermitian matrices and with and . If a solution exists with achieved objective value for the LMIs,

and


or

then the scaled -gap metric at is bounded above by ,


ETH, April 3, 2008

Simultaneous Scaling of and -gapTheorem 3 At a fixed frequency , consider the frequency response of a plant model

stabilizing controllers and and consider positive definite hermitian matrices and with and . If a solution exists with achieved objective value and for the LMIs,

and


or

Then the scaled at is bounded below by and the scaled -gap metric at is bounded above by

and

! ! !


ETH, April 3, 2008

Step1. Chose parameters andStep2. Let the initial values, and ,

be sufficiently large, and find initial values for and such that, and Then initialize and choose and

such that, andStep3. Compute the analytic centers,

Step4. If do not make scaled and scaled violate the WNC and satisfy the followings,

and or if do not make scaled and

scaled violate the WNC and satisfy the followings,

and then

where

Otherwise,

Step5. Stop, if and . Otherwise and go to Step3.

Simultaneous Scaling Algorithm


ETH, April 3, 2008

Numerical Example

Plant model,

- stabilizing controller

- stabilizing controller

and we want to make larger the guaranteed lower bound on


ETH, April 3, 2008

Numerical Example

Scalings for maximizing

Simultaneous scalings increased the generalized stability margin and decreased the -gap


experimental certification of jet engine...

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