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Experimental Certification of Jet Engine...
Transcript of 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
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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)
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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
Tuesday, April 3, 2012
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
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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
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bP,C ! [0, 1]
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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.
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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
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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
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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
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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, ")
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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.
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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
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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
! !
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arcsin !!(Ci, Cj) ! arcsin" " arcsin bP,Ci
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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"
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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
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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
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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
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Numerical Examples (stability certification)
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
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Numerical Examples (stability certification)
Iteration for stability certification After 31 tests, stability certification for 60 controllers (blue circles) was
completed.
arcsin !!(Ci, Cj) < arcsin bP,Ci
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Numerical Examples (margin certification)
27 controllers certified as , by 7 experiments.
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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
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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
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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
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Simultaneous Scalings
For an efficient controller certification, larger and smaller should be incorporated if possible
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Simultaneous Scalings
For an efficient controller certification, larger and smaller should be incorporated if possible
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Simultaneous Scalings
For an efficient controller certification, larger and smaller should be incorporated if possible
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Simultaneous Scalings
For an efficient controller certification, larger and smaller should be incorporated if possible
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Problem Setting
, positive definite symmetric matrices, constant over frequency
, positive definite hermitian matrices at a fixed frequency
! ! !
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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)
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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%
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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
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Implementation of Controller Certification
Development of GUI in corporation with SC Solutions
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I. Controller CertificationII. Estimation of the Generalized Stability
Margin
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I. Controller CertificationII. Estimation of the Generalized Stability
Margin
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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
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Estimating Frequency Response
Estimating one frequency response value - SISO case Input / Output
Model
DFTs
Empirical Transfer Function Estimate
SISO Error Analysis
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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
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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
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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
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Single Frequency Result to Neighboring Frequencies Bound
In order to bound the error between two fixed frequency points Interpolation error and dependencies
with and
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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
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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
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Numerical Example
Matrix DFT:
The maximum singular value plot of a) and b)ETFE of
a)
b)
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Numerical Example
Effect of different length of pre-experiments,
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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
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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)
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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
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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
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Certification for JSF engine
DoD SBIR
F135 test stand
Test data
Acknowledgment
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Certification for JSF engine
DoD SBIR
F135 test stand $Test data
Acknowledgment
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Certification for JSF engine
DoD SBIR
F135 test stand $
$
Test data
Acknowledgment
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Certification for JSF engine
DoD SBIR
F135 test stand $
$¢
Test data
Acknowledgment
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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
! ! !
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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
! ! !
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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
! ! !
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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
! ! !
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Numerical Example
Scalings for maximizing
Simultaneous scalings increased the generalized stability margin and decreased the -gap
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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
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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
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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
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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
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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
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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
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Scaled -gap
Using and , minimization of the scaled -gap can be formulated as
minimize
subject to
Scaled -gap metric
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Scaled -gap
Using and , minimization of the scaled -gap can be formulated as
minimize
subject to
Scaled -gap metric
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LMI Formulation for Scaling -gap Using Schur complement, Minimize
subject to
Minimize
subject to
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LMI Formulation for Scaling -gap Using Schur complement, Minimize
subject to
Minimize
subject to
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LMI Formulation for Scaling -gap Using Schur complement, Minimize
subject to
Minimize
subject to
Iterative minimization on and
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Define the following matrices and ,
then define two sets and ,
LMI Formulation for Scaling -gap
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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
with following properties,
or
then the scaled -gap metric at is bounded above by ,
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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
with following properties,
or
then the scaled -gap metric at is bounded above by ,
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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
with following properties,
or
then the scaled -gap metric at is bounded above by ,
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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
! ! !
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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
! ! !
Tuesday, April 3, 2012
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
! ! !
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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
! ! !
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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
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Numerical Example
Plant model,
- stabilizing controller
- stabilizing controller
and we want to make larger the guaranteed lower bound on
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Numerical Example
Scalings for maximizing
Simultaneous scalings increased the generalized stability margin and decreased the -gap
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