Estimate Initial Conditions for Simulating Identified Models
When you simulate a dynamic system that does not start from rest, you must initialize that simulation to match the starting conditions. For example, you may be comparing your simulated response to measured data, or to a previous simulation or computation. You might want to start your simulation at a steadystate condition, or continue a simulation from a previous simulation.
Summary of InitialCondition Estimation Approaches
System Identification Toolbox™ provides multiple techniques for determining initial conditions (ICs) to use. These techniques include using:
Model identification results
Current or past input/output data measurements
State information from previous simulations
The following table summarizes what approaches are available and when to use them. For conciseness, this table uses the term "simulating" broadly to mean "simulating or predicting." For more information, see Simulate and Predict Identified Model Output.
Approaches for Estimating Initial Conditions
When to Use  Approach  Method Summary 

You are simulating an idss ,idtf , idproc ,
idpoly , or
idgrey model
response using the same input data with which you identified
your model.  Extract ICs from model identification results (returned values or estimation report). Estimation returns IC information in one of the following forms.

Examples: Resolve Fit Value Differences Between Model Identification and compare Command Apply Initial Conditions When Simulating Identified Linear Models 
You want to reproduce the results generated by compare (or
generated by the Model Output plot in the
app).  Use returned ICs from

Examples: Match Output of compare Command and Nonlinear ARX Model Block Apply Initial Conditions When Simulating Identified Linear Models 
You are simulating model response to a given input signal, and want to start the simulation at a point that leads to the closest match to a given output signal.  Use Use
If the original model is

Examples: Apply Initial Conditions When Simulating Identified Linear Models Estimate Initial Conditions for Linear Model Using Measured Data 
You are continuing a simulation, such as for online
processing. You are using a split data set, such as for nlarx model
when findstates is too computationally
expensive.  data2state
If original model is idtf , idproc , or
idpoly , convert to statespace form
first using idss .
You can also use data2state implicitly by specifying
your past data as a proxy for initial conditions when using
sim or predict with any
dynamic model.For more information, see Estimate Initial Conditions from Measured Data.  sys_ss =
idss(sys) x0 =
data2state(sys,z_p) or IO
=
struct or
opt = predictOptions
('InitialCondition',IO) .
examples: Estimate Initial Conditions for a Nonlinear ARX Model, Understand Use of Historical Data for Model Simulation 
You are working with a continuing simulation, and want to
base initial conditions for the current simulation segment on
the previous segment. You intend to start your main simulation in steady state, and you want to have a prequel which simulates the runup to steady state from rest.  Use state information from previous simulation or
presimulation. If original model is idtf , idproc ,
or idpoly , convert to statespace form
first using idss .
For more information, see Use Initial Conditions from Past State Information.  sys_ss =
idss(sys) x0 =
x(end) from [y,y_sd,x] =
sim(sys_ss,z_p) example: Continue Simulation of an Identified Model using Past States 
InitialCondition Estimation Techniques
The preceding table calls out various techniques for initialcondition estimation. The following sections summarize these techniques by the estimation data source.
Estimate Initial Conditions from Measured Data
Estimate the initial conditions from the measurement data from which you are getting your simulation inputs. Techniques include the following.
Extract initial conditions from modelidentification results, such as from the identification report for your model or as returned values from
compare
. X_{0} represents an initial state vector and IC represents aninitialCondition
object. For examples, see Resolve Fit Value Differences Between Model Identification and compare Command, Match Output of compare Command and Nonlinear ARX Model Block, and Apply Initial Conditions When Simulating Identified Linear Models.Estimate initial conditions using measured input/output data, using the function
findstates
to estimate the initial states consistent with the data set. For an example, see Estimate Initial Conditions for Linear Model Using Measured Data.
Estimate Initial Conditions from Past Measurement Data
Estimate the initial conditions from the measurement data immediately preceding your simulation start. Techniques include the following.
Using the function
data2state
to estimate that data set end state. The past end state then becomes the initial state for followon simulation. For an example, see Estimate Initial Conditions for a Nonlinear ARX Model.Using your past data as a proxy for initial conditions when using
sim
orpredict
. When you run your simulation, the model internally callsdata2state
for you. For an example, see Understand Use of Historical Data for Model Simulation.Splitting the data set you are using into two parts. Make the first part just large enough to allow
data2state
to calculate an accurate end state. This splitdata approach can be much less computationally intensive than usingfindstates
on the full data set, especially for nonlinear ARX models. For an example, see Estimate Initial Conditions for a Nonlinear ARX Model.
Use Initial Conditions from Past State Information
Use state information from previous simulations or presimulations. If no previous simulation is available, you can presimulate from rest or a known initial condition up to the point of your simulation start. The end states of the past or presimulation are the starting states for the simulation you want to perform. For an example, see Continue Simulation of an Identified Model using Past States.
Estimate Initial Conditions for Simulating Linear Models
The findstates
and data2state
functions
are the most general tools for determining initial or end states from any
measurements of input/output data. However, to use these functions, you must have a
model structure that includes explicit state representation. Linear models other
than idss
and idgrey
do not use explicit
states. To find the equivalent initial states, you can convert these models to
idss
models and find the initial states from the
idss
versions. As an alternative to
findstates
, you can also use compare
to obtain initial conditions without converting your models into statespace form.
For these models, compare
returns an initialCondition
object that contains information on the model free
response, in statespace form, and the corresponding initial states.
After you estimate the initial conditions, you can incorporate them into your commandline simulation syntax, your Simulink^{®} model, or the System Identification app.
The following examples illustrate initialcondition estimation for linear models.
The first example uses findstates
and
compare
. The second and third examples use historical data.
That is, the examples determine initial conditions from the data immediately
preceding the simulation start time.
Estimate Initial Conditions for Linear Model Using Measured Data
Use different methods to obtain initial conditions for a linear model and compare the results.
Estimate a transfer function for measured data. Return the initial conditions in ic
.
load iddata1 z1; [sys,ic] = tfest(z1,3); ic
ic = initialCondition with properties: A: [3x3 double] X0: [3x1 double] C: [0.2303 5.9117 2.2283] Ts: 0
ic.X0
ans = 3×1
1.7569
2.6195
6.5177
sys
is a continuoustime identified transfer function (idtf
) model. ic
is an initialCondition
object. An initialCondition
object represents the free response of the system in statespace form. The object includes the converted A
and C
statespace matrices that correspond to sys
and the estimated initial state vector X0
.
You can also use findstates
to find initial conditions for linear models, but only if they are in idss
or idgrey
form.
Convert sys
to statespace form.
sys_ss = idss(sys);
Use findstates
to estimate the initial states.
X0 = findstates(sys_ss,z1)
X0 = 3×1
1.7569
2.6195
6.5177
Now use compare
to estimate the initial states, using the original transfer function model.
[y_tf,fit,ic2] = compare(z1,sys); ic2
ic2 = initialCondition with properties: A: [3x3 double] X0: [3x1 double] C: [0.2303 5.9117 2.2283] Ts: 0
ic2.X0
ans = 3×1
1.7569
2.6195
6.5177
The initial state vectors are identical in all three cases.
Understand Use of Historical Data for Model Simulation
Use historical inputoutput data as a proxy for initial conditions when simulating your model. You first simulate using the sim
command and specify the historical data using the simOptions
option set. You then reproduce the simulated output by manually mapping the historical data to initial states.
Load a twoinput, oneoutput data set.
load iddata7 z7
Identify a fifthorder statespace model using the data.
sys = n4sid(z7,5);
Split the data set into two parts.
zA = z7(1:15); zB = z7(16:end);
Simulate the model using the input signal in zB
.
uSim = zB;
Simulation requires initial conditions. The signal values in zA
are the historical data, that is, they are the input and output values for the time immediately preceding data in zB
. Use zA
as a proxy for the required initial conditions.
IO = struct('Input',zA.InputData,'Output',zA.OutputData); opt = simOptions('InitialCondition',IO);
Simulate the model.
ysim = sim(sys,uSim,opt);
Now reproduce the output by manually mapping the historical data to initial states of sys
. To do so, use the data2state
command.
xf = data2state(sys,zA);
xf
contains the state values of sys
at the time instant immediately after the most recent data sample in zA
.
Simulate the system using xf
as the initial states.
opt2 = simOptions('InitialCondition',xf);
ysim2 = sim(sys,uSim,opt2);
Plot the output of the sim
command ysim
and the manually computed results ysim
2.
plot(ysim,'b',ysim2,'r')
ysim2
is the same as ysim
.
Continue Simulation of an Identified Model using Past States
Estimate the initial conditions for a simulation segment of an identified model by using the end states of the previous simulation segment.
Load data z1, which is an iddata
object containing input and output data. Estimate a fifthorder linear model from the data.
load iddata1 z1 plot(z1)
sys = n4sid(z1,5);
Divide the data into two segments. One segment z1past
represents data for a past simulation, and starts at rest. The second segment z1next
represents data for a followon simulation.
z1past = z1(1:150); z1next = z1(150:300);
Simulate the response to the input data in z1next
without specifying initial conditions.
ynext_no_ic = sim(z1next,sys);
plot(ynext_no_ic,z1next)
legend
title('Simulation Continuation Without Setting ICs')
Now determine the end states, x0
, for the response to z1past
input.
[ypast,y_sd,xpast] = sim(z1past,sys); x0 = xpast(end,:)';
Use the end states to specify the initial states for the followon response to z1next
.
opt = simOptions('InitialCondition',x0); ynext_ic = sim(z1next,sys,opt); figure plot(ynext_ic,z1next) legend title('Simulation Continuation with Sychronized ICs')
Estimate Initial Conditions for Simulating HammersteinWiener and Nonlinear GreyBox Models
When you are simulating a HammersteinWiener or a nonlinear greybox model and
want to use initial conditions consistent with a measurement data set
z
, use findstates
.
Simulate HammersteinWiener Model in Simulink
Compare the simulated output of a HammersteinWiener Model block to the measured output of a system. You improve the agreement between the measured and simulated responses by estimating initial state values.
Load the sample data.
load twotankdata
Create an iddata
object from the sample data. Set 'Tstart'
to 0 so that the data start time matches the Simulink start time 0f 0 s.
z1 = iddata(y,u,0.2,'Tstart',0,'Name','Two tank system');
Estimate a HammersteinWiener model using the data.
mw1 = nlhw(z1,[1 5 3],idPiecewiseLinear,idPiecewiseLinear);
You can now simulate the output of the estimated model in Simulink using the input data in z1
. To do so, open a preconfigured Simulink model.
model = 'ex_idnlhw_block';
open_system(model);
The model uses the Iddata Source, HammersteinWiener Model, and Scope blocks. The following block parameters have been preconfigured to specify the estimation data, estimated model, and initial conditions:
Block parameters of Iddata Source block:
IDDATA Object 
z1
Block parameters of HammersteinWiener Model block:
Model 
mw1
Initial conditions 
Zero
(default)
Run the simulation.
View the difference between measured output and model output by using the Scope block.
simOut = sim(model);
open_system([model '/Scope'])
To improve the agreement between the measured and simulated responses, estimate an initial state vector for the model from the estimation data z1
, using findstates
. Specify the maximum number of iterations for estimation as 100. Specify the prediction horizon as Inf
, so that the algorithm computes the initial states that minimize the simulation error.
opt = findstatesOptions; opt.SearchOptions.MaxIterations = 100; x0 = findstates(mw1,z1,Inf,opt);
Set the Initial conditions block parameter value of the HammersteinWiener Model block to State Values
. The default initial states are x0
.
set_param([model '/HammersteinWiener Model'],'IC','State values');
Run the simulation again, and view the difference between measured output and model output in the Scope block. The difference between the measured and simulated responses is now reduced.
simOut = sim(model);
Estimate Initial Conditions for Simulating Nonlinear ARX Models
When you are simulating a nonlinear ARX model and want to use initial conditions
consistent with a measurement data set z
, you have a choice for
which approach to take.
You can use
findstates
. For nonlinear ARX models however, this approach has the cost of being computationally expensive if the data set is large.X0 = findstates(m,z,Inf);
You can break the data set into two portions. Use the first
nx
samples of the input/output data as past data, wherenx
is the largest regressor delay in the model. Then usedata2state
to compute the end state of the "past data." You are then able to simulate the model for the remainingnx+1:end
input data with whatever simulation approach you choose. Bothcompare
and the System Identification app use this approach automatically for nonlinear ARX models.nx = max(getDelayInfo(m)); % Find the largest regressor delay past_data = z1(1:nx); X0 = data2state(mw1,z1(1:nx));
The following example illustrates the use of splitdata technique for a nonlinear ARX model.
Estimate Initial Conditions for a Nonlinear ARX Model
Use two different techniques to estimate the initial conditions for a nonlinear ARX model. Simulate each alongside the measurement data, and compare with a simulation that initializes at rest.
Load the input/output data z
and plot it. Use the first 1000 points of z
to estimate a nonlinear ARX model.
load twotankdata z = iddata(y,u,0.2); z1 = z(1:1000); plot(z1) title('Estimation data')
mdlnlarx = nlarx(z1,[5 1 3],idWaveletNetwork);
Extract the first 250 points of z1 (50 seconds) as z1sim
for comparison simulations. Simulate mdlarx
without setting initial conditions, and plot the response alongside the measured output in z1sim
.
z1sim = z1(1:250); data_no_ic = sim(mdlnlarx,z1sim); plot(data_no_ic,z1sim) title('Nonlinear ARX Model with No Initial Condition Setting') grid on legend('location','se')
Use findstates
to estimate the initial conditions. In order to minimize the simulation error, specify Inf
for the prediction horizon. Then set the sim
option for InitialCondition
to the findstates
results. Plot the response alongside the measured output.
x0fs = findstates(mdlnlarx,z1sim,Inf); opt = simOptions('InitialCondition',x0fs); data_fs = sim(mdlnlarx,z1sim,opt); figure plot(data_fs,z1sim) title('Nonlinear ARX Model with IC Estimation Using findstates') grid on legend('location','se')
The response and the measured output now start at roughly the same point.
Now use the data2states
method to estimate the initial conditions. First, break the data set into two portions. Use the first nx
samples of z1sim
as "past data", where nx
is the largest regressor delay in the model.
nx = max(getDelayInfo(mdlnlarx))
nx = 5
z1past = z1sim(1:nx); z1sim2 = z1sim(nx+1:end);
Use data2state
to compute the end state of the "past data." Simulate the response using input starting at z1sim(nx+1)
. Plot the response alongside the full z1sim
measurement data so that you can compare with the findstates
plot.
x0d2s = data2state(mdlnlarx,z1past); opt = simOptions('InitialCondition',x0d2s); data_d2s = sim(mdlnlarx,z1sim2,opt); figure plot(data_d2s,z1sim) title('Nonlinear ARX Model with IC Estimation Using data2state') grid on legend('location','se')
If you executed each section separately, you may find that the data2states
method completed more quickly than the findstates
method.
Now compare the responses for all three initialcondition cases.
plot(data_fs,data_d2s,data_no_ic) title('Nonlinear ARX model Responses for Three IC Approaches Compared') legend('location','se')
The responses for cases using findstates
and data2state
are virtually the same. The response for the case where the initial conditions were not set converges eventually, but not until after 30 seconds.
See Also
compare
 compareOptions
 sim
 simOptions
 predict
 predictOptions
 findstates
 data2state