ssInterpolant
Syntax
Description
For a collection of local ss
models sampled in time or parameter space,
ssInterpolant
builds a linear time-varying (LTV) or linear
parameter-varying (LPV) model that interpolates local LTI behaviors into global LTV or LPV
behavior. You can also use ssInterpolant
to turn analytic LTV or LPV
models into gridded LTV or LPV models, or to resample gridded LTV or LPV
models.
constructs the LTV or LPV interpolant from the array of state-space models
sys
= ssInterpolant(ssArray
)ssArray
. For this array, the ssArray.SamplingGrid
property specified the underlying time or parameter grid, which can be rectangular or
consist of scattered samples, and ssArray.Offsets
specifies the
linearization offsets. (since R2024a)
Examples
Create LPV Model from Batch Linearization Results
This example shows how to create an LPV model from an array of state space models and associated offsets obtained from batch linearization of a water-tank Simulink® model.
Open the Simulink model.
mdl = 'watertankNLModel';
open_system(mdl)
Specify the initial condition for water height.
h0 = 10;
Specify model linear analysis points.
io(1) = linio('watertankNLModel/Step',1,'input'); io(2) = linio('watertankNLModel/H',1,'output');
Simulate the model and extract operating points at time snapshots.
tlin = [0 30 40 50 60 70 80]; op = findop(mdl,tlin);
Compute the linearizations along with offsets.
options = linearizeOptions('StoreOffsets',true);
[linsys,~,info] = linearize(mdl,io,op,options);
Specify a grid of height values and construct the LPV model. For this model, height is the first state.
h = zeros(size(tlin)); for ct=1:numel(h) h(ct) = op(ct).States(1).x; end linsys.SamplingGrid.H = h; linsys.Offsets = info.Offsets;
gLPV = ssInterpolant(linsys,'spline')
Continuous-time state-space LPV model with 1 outputs, 1 inputs, 1 states, and 1 parameters.
Simulate the watertank
model.
h0 = 15; sim(mdl); usim = getElement(logsOut,'V'); hsim = getElement(logsOut,'H');
Simulate the LPV model with the exact parameter trajectory.
t = linspace(0,150,200); p = @(t,x,u) x(1); u = interp1(usim.Values.Time,usim.Values.Data,t); ylpv = lsim(gLPV,u,t,h0,p);
Plot the results.
plot(hsim.Values.Time,hsim.Values.Data,t,ylpv,'--') legend('Simulink model','LPV model')
The LPV model provides a good approximation of the nonlinear model.
Convert Analytic LPV Model to Gridded LPV Model
For this example, dataFcnMaglev.m
defines the matrices and offsets of a magnetic levitation system. The magnetic levitation controls the height of a levitating ball using a coil current that creates a magnetic force on the ball.
Create an LPV model.
h0 = 1;
lpvSys = lpvss('h',@dataFcnMaglev,0,0,h0)
Continuous-time state-space LPV model with 1 outputs, 1 inputs, 2 states, and 1 parameters.
lpvSys.StateName = {'h','hdot'}; lpvSys.InputName = 'current'; lpvSys.InputName = 'height';
Define parameter variation for height h
.
hmin = 0.05; hmax = 0.25; hcd = linspace(hmin,hmax,5); SamplingGrid.h = hcd;
Convert to a gridded model.
glpvSys1 = ssInterpolant(lpvSys,SamplingGrid)
Continuous-time state-space LPV model with 1 outputs, 1 inputs, 2 states, and 1 parameters.
This command is equivalent to constructing a gridded model from an array of local models sampled at the same parameter space.
ssArray = psample(lpvSys,SamplingGrid); glpvSys2 = ssInterpolant(ssArray);
Compare Gridded LPV Model Extrapolation Methods
For this example consider an array of state space models and associated offsets obtained from batch linearization of a water-tank Simulink® model in the Create LPV Model from Batch Linearization Results example.
The dynamics of this water-tank system are described by this equation.
Load the linearization results.
load watertankLinModel.mat
Construct a gridded model with the spline
interpolation method and clip
extrapolation method.
glpv1 = ssInterpolant(linsys,info.Offsets,"spline","clip");
Construct a gridded model with the spline
extrapolation.
glpv2 = ssInterpolant(linsys,info.Offsets,"spline","spline");
Compare simulation results of both gridded models when the initial height starts from a point outside the constructing grid. Set the height such that the tank is almost empty.
h0 = 0.4; sim("watertankNLModel.slx"); usim = getElement(logsOut,"V"); hsim = getElement(logsOut,"H");
Simulate the LPV models with the exact parameter trajectory.
t = linspace(0,150,200); p = @(t,x,u) x(1); u = interp1(usim.Values.Time,usim.Values.Data,t); ylpv1 = lsim(glpv1,u,t,h0,p); ylpv2 = lsim(glpv2,u,t,h0,p); plot(t,ylpv1,t,ylpv2,"--",hsim.Values.Time,hsim.Values.Data,".-") legend("clip extrapolation","spline extrapolation", ... "nonlinear simulation",Location="best")
When the tank is almost empty, the spline
method captures the rapid slope change resulting from the nonlinearity better than the clip
method.
Input Arguments
ssArray
— Array of state-space models
array of ss
objects
Array of state-space (ss
) models. Use the
SamplingGrid
property of the state-space object to specify the
time and parameter dependence of the models in ssArray
, and assign
time and parameter values to each model. The field names of
SamplingGrid
become the parameters of the LPV model, except for
the field Time
which is interpreted as describing the time
dependence. If Time
is the only field,
ssInterpolant
returns an LTV model. The function ignores fields
with constant value.
Additionally, you can use the
Offsets
property of ssArray
to specify the
derivative, state, input, and output linearization offsets (since R2024a). See ss
reference page for more information on the
SamplingGrid
and Offsets
properties.
You can also obtain ssArray
and corresponding
offsets
in one of the following ways:
Perform batch linearization of a Simulink® model with
StoreOffsets
option. For more information, seelinearize
(Simulink Control Design) andlinearizeOptions
(Simulink Control Design).Sample an existing LTV or LPV model. For more information, see
sample
.
offsets
— Model offsets
structure array
Model offsets, specified as a structure array with the same dimensions as
ssArray
. Each offset structure must have the following
fields:
Field | Description |
---|---|
x | State offsets, specified as a column vector of length
nx, where
nx is the number of states in
ssArray |
dx | Derivative offsets, specified as a column vector of length
nx, where
nx is the number of states in
ssArray |
u | Input offsets, specified as a column vector of length
nu, where
nu is the number of inputs in
ssArray |
y | Output offsets, specified as a column vector of length
ny, where
ny is the number of outputs in
ssArray |
You can also obtain ssArray
and corresponding
offsets
in one of the following ways:
Perform batch linearization of a Simulink model with
StoreOffsets
option. For more information, seelinearize
(Simulink Control Design) andlinearizeOptions
(Simulink Control Design).Sample an existing LTV or LPV model. For more information, see
sample
.
You can also set any of the fields to []
when the model has no
offsets.
vSys
— LTV or LPV model
ltvss
object | lpvss
object
LTV or LPV model, specified as an ltvss
or lpvss
object.
If vSys
is an ltvss
or lpvss
object, ssInterpolant
builds a gridded LTV or LPV model at the grid
values specified using S
.
S
— Sampling grid
struct
Sampling grid values, specified as a structure. S
defines a
grid of time and/or parameter values where vSys
is sampled. The
gridded LTV or LPV model interpolates between these grid points.
For LTV models,
S
is a structure array with fieldTime
. For example, if an arrayTvalues
specifies the sampling time,S
is the following structure.S = struct('Time',Tvalues)
For LPV models,
S
is a structure with fields for parameter names corresponding to theParameterName
property of the LPV modelvSys
, andTime
. For example, to create a rectangular grid, you can usendgrid
to generate values for this structure. For example, ifvSys
has two parameters named'speed'
and'altitude'
with values specified using arraysP1vals
andP2vals
, respectively,S
is the following structure.[P1,P2] = ndgrid(P1vals,P2vals); S = struct('speed',P1,'altitude',P2)
If the dynamics of your LPV model do not have an explicit dependence on time, do not specify the field
Time
forS
.
In discrete time, specify Time
as integer index values
k
that count the number of sampling periods Ts
.
The absolute time is given by t
= k*Ts
.
IMethod
— Interpolation method
'linear'
(default) | 'nearest'
| 'next'
| 'previous'
| 'pchip'
| 'cubic'
| 'spline'
| 'makima'
| 'natural'
Interpolation method, specified as one of these depending on the type of parameter grid.
Rectangular grids —
'linear'
,'nearest'
,'next'
,'previous'
,'pchip'
,'cubic'
,'spline'
, or'makima'
. SeegriddedInterpolant
for more information.Scattered grids —
'linear'
,'nearest'
, or'natural'
. SeescatteredInterpolant
for more information.
If you omit both the IMethod
and EMethod
arguments, the values default to 'linear'
for
IMethod
and 'clip'
for
EMethod
.
EMethod
— Extrapolation method
'clip'
(default) | 'linear'
| 'nearest'
| 'next'
| 'previous'
| 'pchip'
| 'cubic'
| 'spline'
| 'makima'
| 'none'
Extrapolation method, specified as one of these depending on the type of parameter
grid. 'clip'
, 'linear'
,
'nearest'
, 'next'
,
'previous'
, 'pchip'
, 'cubic'
,
'spline'
, or 'makima'
. In addition, you can
specify 'none'
if you want queries outside the domain of your grid to
return NaN
values.
Rectangular grids —
'clip'
,'linear'
,'nearest'
,'next'
,'previous'
,'pchip'
,'cubic'
,'spline'
, or'makima'
. In addition, you can specify'none'
if you want queries outside the domain of your grid to returnNaN
values. SeegriddedInterpolant
for more information.Scattered grids —
'linear'
,'nearest'
, or'none'
. SeescatteredInterpolant
for more information.
If you omit both the IMethod
and EMethod
arguments, the values default to 'linear'
for
IMethod
and 'clip'
for
EMethod
.
The 'clip'
method limits each parameter values to its
[pmin,pmax]
range in the grid. This method is similar to the 'Clip'
method in
Simulink lookup table blocks.
Output Arguments
sys
— Gridded LTV or LPV model
GriddedLTVSS
object | GriddedLPVSS
object
Gridded LTV or LPV model, returned as a GriddedLTVSS
or
GriddedLPVSS
object, respectively.
In addition to the properties in ltvss
and lpvss
objects, the GriddedLTVSS
and GriddedLPVSS
objects
contain the properties Grid
, Interpolation
,
and Extrapolation
, that specify the grid values, interpolation
method, and extrapolation method, respectively, used to construct the gridded
model.
Limitations
Scattered interpolation (since R2023b) is supported only for parameter grids up to three dimensions.
Version History
Introduced in R2023aR2024a: Use new syntax to create gridded models when offsets are stored in ss
array
Use the new syntax sys = ssInterpolant(ssArray)
to construct a
gridded model when offsets are stored in the Offsets
property of the
state-space model array ssArray
.
R2023b: Support for scattered parameter samples
Starting in R2023b, ssInterpolant
can interpolate state-space data
specified at scattered points in 2-D or 3-D parameter space. The maximum of three parameters
includes time values when the LPV model explicitly depends on time.
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