Model Reduction Software
Name
emgr  Empirical Gramian framework for model reduction of inputoutput systems.
Synopsis
Empirical gramians can be computed for linear and nonlinear statespace control systems for purposes of model order reduction (MOR), system identification (SYSID) and uncertainty quantification (UQ).
Model reduction using empirical gramians can be applied to the statespace, to the parameterspace or both through combined reduction.
For state reduction, balanced truncation of the empirical controllability gramian and the empirical observability gramian, or alternatively, direct truncation (approximate balancing) of the empirical cross gramian (or the empirical linear cross gramian for largescale linear systems) is available.
For parameter reduction, parameter identification and sensitivity analysis the empirical sensitivity gramian (controllability of parameters) or the empirical identifiability gramian (observability of parameters) are provided.
Combined state and parameter reduction is enabled by the empirical joint gramian, which computes controllability and observability of states (cross gramian) and observability of parameters (crossidentifiability gramian) concurrently.
The empirical gramian framework  emgr
is a compact opensource toolbox for (empirical) GRAMIANbased model reduction and compatible with OCTAVE and MATLAB.
emgr
provides a common interface for the computation of empirical gramians and empirical covariance matrices.
Scope
 Model Reduction
 Parametric Model Order Reduction (pMOR)
 Nonlinear Model Order Reduction (nMOR)
 Robust Reduction
 Sensitivity Analysis
 Parameter Identification
 Parameter Reduction
 Combined State and Parameter Reduction
 Decentralized Control
 Optimal Sensor Placement
 Optimal Actuator Placement
 Linear & Nonlinear Control Systems
 Time Invariant & Time Varying
 Parametrized  Parametric Systems
 Discretized PDEs
 Systems with:
 Vector Field f: ẋ(t) = f(x(t),u(t),p,t)
 Output Functional g: y(t) = g(x(t),u(t),p,t)
Download
Get emgr
here: emgr.m (Version: 5.0)
[mirror]
[source]
[meta]
[emgr_oct.m]
[emgr_legacy.m]
^{(emgr is written in the matlab programming language and requires Octave (>= 4.0) or Matlab (>= 2016b). emgr has no dependencies on other toolboxes or packages.)}
License
All source code is licensed under the open source BSD 2clause license:
Copyright (c) 20132016, Christian Himpe
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
1. Redistributions of source code must retain the above copyright notice, this
list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice,
this list of conditions and the following disclaimer in the documentation
and/or other materials provided with the distribution.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
Usage
General Usage: W = emgr(f,g,s,t,w,pr,nf,ut,us,xs,um,xm,pm);
Minimal Usage: W = emgr(f,g,s,t,w);
About Info Usage: v = emgr('version');
Arguments
f
 Handle to a function with signaturex = f(x,u,p,t)
, the system's vector field.g
 Handle to a function with signaturey = g(x,u,p,t)
, the output functional.g = 1
impliesy = x
.s
 Three component vectors = [M,N,Q]
holding number of inputs, states and outputs.t
 Two component vectort = [h,T]
holding time step and stop time.w
 A character selecting the gramian type, for details see Gramians.
Optional Arguments
pr
 System's parameters (Default:0
;'s'
,'i'
,'j'
require two columns for min and max parameter),vector
 Column vector holding the parameters,matrix
 Set of column vectors holding different sets of parameters.
nf
 Ten component vector holding options; (Default:0
), for details see Option Flags.ut
 Input time series (Default:1
),handle
 Handle to function with signatureu_t = u(t)
,1
 Delta impulse input,∞
 Linear chirp input using the havercosine.
us
 Steadystate input (Default:0
),scalar
 Sets allM
input components to provided value,vector
 Steadystate input column vector of dimensionM
.
xs
 Steadystate, also used as nominal initial state (Default:0
),scalar
 Sets allN
steadystate components to provided value,vector
 Steadystate column vector of dimensionN
.
um
 Input perturbation scales (Default:1
),scalar
 Set all maximum scales to argument,vector
 Set maximum to scale to argument,matrix
 Set scales to argument, used as is.
xm
 Initial state perturbation scales (Default:1
),scalar
 Set all maximum scales to argument,vector
 Set maximum to scale to argument,matrix
 Set scales to argument, used as is.
pm
 Parameter perturbation scales, RESERVED (Default:[]
).
Gramians
'c'
 Empirical Controllability Gramian (WC
),'o'
 Empirical Observability Gramian (WO
),'x'
 Empirical Cross Gramian (WX
akaWCO
orXCG
),'y'
 Empirical Linear Cross Gramian (WY
),'s'
 Empirical Sensitivity Gramian (WS
),'i'
 Empirical Identifiability Gramian (WI
),'j'
 Empirical Joint Gramian (WJ
).
WZ
can be selected by option nf(7)=1
and is compatible with the empirical cross gramian WX
, the empirical linear cross gramian WY
and the empirical joint gramian WJ
.
Option Flags
nf(1)
 Center timer series around (Default:0
),= 0
 Zero,= 1
 Initial state,= 2
 Steadystate (assumed to be reached at final time),= 3
 Arithmetic average over time,= 4
 Rootmeansquare over time.= 5
 Midrange over time.
nf(2)
 Input scale sequence (Default:0
),= 0
 Singleum = um
;= 1
 Linearum = um*[0.25,0.50,0.75,1.0]
;= 2
 Geometricum = um*[0.125,0.25,0.5,1.0]
;= 3
 Logarithmicum = um*[0.001,0.01,0.1,1.0]
;= 4
 Sparseum = um*[0.38,0.71,0.92,1.0]
;
nf(3)
 State scale sequence (Default:0
),= 0
 Singlexm = xm
;= 1
 Linearxm = xm*[0.25,0.50,0.75,1.0]
;= 2
 Geometricxm = xm*[0.125,0.25,0.5,1.0]
;= 3
 Logarithmicxm = xm*[0.001,0.01,0.1,1.0]
;= 4
 Sparsexm = xm*[0.38,0.71,0.92,1.0]
;
nf(4)
 Input transformations (Default:0
),= 0
 Unitum = [um,um]
;= 1
 Singleum = um
;
nf(5)
 State transformations (Default:0
),= 0
 Unitxm = [xm,xm]
;= 1
 Singlexm = xm
;
nf(6)
 Scaled run (Default:0
),= 0
 Unscaled,= 1
 Double run, scale with gramian diagonal,= 2
 Scaled run, scale with steadystate (input).
nf(7)
 NonSymmetric Cross Gramian, onlyWX, WY, WJ
(Default:0
),= 0
 Regular cross gramian,= 1
 Nonsymmetric cross gramian (cross operatorWZ
) for nonsquare, nonsymmetric or nongradient systems.
nf(8)
 Enable nominal input during parameter and state perturbations, onlyWO, WX, WS, WI, WJ
(Default:0
),= 0
 No extra input,= 1
 Extra input for parameter perturbations.= 2
 Extra input for state perturbations.= 3
 Extra input for parameter and state perturbations.
nf(9)
 Center parameter scales, onlyWS, WI, WJ
(Default:0
),= 0
 No centering,= 1
 Center around arithmetic mean,= 2
 Center around geometric mean.
nf(10)
 Detailed or approximate Schurcomplement, onlyWI, WJ
(Default:0
),= 0
 detailed Schurcomplement,= 1
 approximate Schurcomplement.
Configuration
 Custom Solver (global variable:
ODE
) Default solver: Explicit singlestep method SSP32, the optimal three stage, second order strong stability preserving (SSP) RungeKutta integrator.
 Function signature:
y = solver(f,g,t,x0,u,p)
f
 Handle to a function with signaturex = f(x,u,p,t)
, the system's vector field,g
 Handle to a function with signaturey = g(x,u,p,t)
, the output functional.g = 1
impliesy = x
,t
 Three component vectort = [h,T]
holding time step and time stop,x0
 Column vector of dimensionN
holding initial state,u
 handle to function with signatureu_t = u(t)
,p
 Column vector holding parameters.
 Custom Kernel (global variable:
DOT
) Default kernel: Linear L2 kernel, Euclidean inner product.
 Function signature:
w = kernel(x,y)
x
 Discrete time series of dimensionN x (T/h)
,y
 Discrete time series of dimension(T/h) x N
.
 Distributed Empirical Cross Gramian (global variable:
DWX
) Two component vector:
DWX(1)
 Number of columns per partition,DWX(2)
 Index of partition.
 Two component vector:
Extra
 Balancer: balance_co.m (SVDbased balancing method)
 Solver: mysolver.m (Sample custom ODE solver)
 Gauss Kernel: gauss_kernel.m (Gauss kernel)
 Trace PseudoKernel: trace_kernel.m (Trace pseudokernel)
Tests
A set of basic sanity tests is conducted in emgrtest.m, which compute experimental orders of convergence. Further tests are performed on a timeinvariant, linear, statespace symmetric MIMO system, with a negative Lehmer matrixA
and optional linear parametrization:
ẋ = A*x + B*u + p
y = C*x
Demos
Combined Reduction: Nonlinear System (
 
State Reduction: Balanced Gains (
 
Benchmark: Inverse Lyapunov Procedure (
 
Benchmark: Linear Model Reduction (
 
Benchmark: Nonlinear Model Reduction (
 
State Reduction: NonSquare and Nonlinear (
 
Nonlinearity Quantification (
 
Decentralized Control (
 
PDE Reduction: Advection Equation (
 
Nonlinear Second Order Reduction: 5body Choreography (
 
Parameter Identification: Stable Orbits Inside Black Holes (

About
A gramian matrix W
is the result of all inner products of a set of vectors V = (v1 ... vn)
, in other words: W = V' V
.
Properties of (linear) control systems can be assessed by the system gramian matrices, which are based on the controllability and observability operators.
Classically, the controllability gramian and observability gramian are utilized in balancing method.
The cross gramian combines controllability and observability information into a single matrix and thus does not require explicit balancing.
Empirical gramians extend this approach to nonlinear control systems and thus enable nonlinear model reduction.
For linear systems the empirical gramians are equal to the classic gramians.
Yet empirical gramians contain more information about the underlying system; and the empirical cross gramian conveys even additional information.
This makes empirical gramians a versatile tool for mathematical engineering.
The Discrete Empirical Cross Gramian encloses information on the inputoutput behavior of the associated control system as well as approximate Hankel Singular Values and can be computed very efficiently. For extremescale systems, the Linear Cross Gramian, related to Balanced POD, can be utilized. And for parametrized systems, the Joint Gramian, derived from the cross gramian, is available for combined reduction. In case of custom input, an empirical covariance matrix can also be computed.
References
 C. Himpe, M. Ohlberger; "CrossGramianBased Model Reduction: A Comparison"; Submitted, 2016; preprint, source, runmycode
 OneSentence Abstract: Summary and tests for Sylvesterequationbased cross gramian, empirical linear cross gramian and empirical cross gramian.
 C. Himpe, M. Ohlberger; "A note on the cross gramian for nonsymmetric systems"; System Science and Control Engineering 4(1): 199208, 2016; openaccess, source, runmycode
 OneSentence Abstract: A cross gramian matrix for nonsymmetric, nongradient and nonsquare system.
 C. Himpe, M. Ohlberger; "Accelerating the Computation of Empirical Gramians and Related Methods"; 5th IWMRRF, 2015; selfarchived, source
 OneSentence Abstract: Numerical performance enhancements for empirical gramians.
 C. Himpe, M. Ohlberger; "The Empirical Cross Gramian for Parametrized Nonlinear Systems"; Mathematical Modelling , 8(1): 727728, 2015; source, runmycode
 OneSentence Abstract: Use of the empirical cross gramian for parametric model order reduction by averaging.
 C. Himpe, M. Ohlberger; "CrossGramianBased Combined State and Parameter Reduction for LargeScale Control Systems"; Mathematical Problems in Engineering 2014: 113, 2014; openaccess, source, runmycode
 OneSentence Abstract: Introduction of the empirical cross gramian and the derived joint gramian as well as gramianbased combined reduction.
 C. Himpe, M. Ohlberger; "Model Reduction for Complex Hyperbolic Networks"; Proceedings of the European Control Conference: 27392743, 2014; preprint, source, runmycode
 OneSentence Abstract: Cross gramian based model reduction of a timevarying nonsymmetric system with application in network cosmology.
 C. Himpe, M. Ohlberger; "A Unified Software Framework for Empirical Gramians"; Journal of Mathematics 2013: 16, 2013; openaccess, source, runmycode
 OneSentence Abstract: Implementation and mathematical background on empirical gramians under a common interface.
User References
 J. Qi, K. Sun, W. Kang; "Optimal PMU Placement for Power System Dynamic State Estimation by Using Empirical Observability Gramian"; IEEE Transactions on Power Systems 30(4): 20412051, 2015.
 J. Qi, W. Huang, K. Sun, W. Kang; "Optimal Placement of Dynamic Var Sources by Using Empirical Controllability Covariance"; IEEE Transactions on Power Systems PP(99): 110, 2016.
Further References for empirical gramians are in the empirical gramian reference list.
Contact
Send feedback to: ch@gramian.de
Cite
 Cite as: C. Himpe (2016). emgr  Empirical Gramian Framework (Version 5.0) [Software]. http://gramian.de
 BibTeX:
@MISC{emgr,author={C. Himpe},title={{emgr}  Empirical Gramian Framework (Version 5.0)},howpublished={\url{http://gramian.de}},year={2016}}
 DOI: 10.5281/zenodo.162135 (Version 5.0)
 Except where otherwise noted, content on this site is licensed under a CC BY 4.0 license.
 Last Change: 20161020
Links
 emgrref.pdf (emgr reference card)
 emgrposter2015.pdf (emgr poster)
 github.com/gramian/emgr (emgr git repository at github)
 orms.mfo.de/project?id=345 (emgr at Oberwolfach References on Mathematical Software)
 swmath.org/software/7554 (emgr at swMATH)
 freshcode.club/projects/emgr (emgr at freshcode)
 modelreduction.org (Model Order Reduction Wiki)
 morepas.org (Model Reduction for Parametrized Systems)
 en.wikibooks.org/wiki/Control_Systems (Control Systems Wikibook)
 git.io/mtips (Octave / Matlab Snippets)
Notes
 emgr can compute the empirical cross gramian columnwise, and thus in parallel on distributed memory systems.
 emgr is not explicitly parallelized but multicore ready by extensive vectorization and implicit parallelization.
 emgr has highlighted loops that qualify for explicit parallelization using
parfor
.  emgr can use GPGPU based heterogeneous computing by automatic offloading through the BLAS backend including hUMA.
 emgr consists of a single file and has less than 500 lines of code!
Troubleshooting
 Issue: An empirical gramian contains very large or infinity values.
 Fix: Most likely, one of the sampled trajectory is not stable, make sure the perturbations (scales) or parameters destabilize the system and a suitable ODE solver is used.
 Issue: The identifiability gramian
WI{2}
or crossidentifiability gramianWJ{2}
are zero. Fix: The steady state (initial state) needs to introduce energy into the system; usually this means
xs
≠ 0 or settingnf(8) = 1
.
 Fix: The steady state (initial state) needs to introduce energy into the system; usually this means
See Also
 Model Reduction Routines (Another Empirical Gramian Software; WC and WO only)
 gram (Octave Control Package; linear WC and linear WO only)