ltvss
Description
Use ltvss
to represent linear state-space models whose dynamics
vary with time.
You can use ltvss
to create continuous-time or discrete-time linear
time-varying state-space models. In continuous time, an ltvss
model is
described by the following state-space equations.
Here, A(t), B(t), C(t), D(t), and E(t) are time-varying state-space matrices and δ0(t), x0(t), u0(t), and y0(t) are time-dependent derivative, state, input, and output offsets, respectively.
In discrete time, an ltvss
model is described by the following
state-space equations.
Here, the integer index k counts the number of sampling periods
Ts
.
You can use an ltvss
object to model:
Linear systems whose coefficients vary in time.
Nonlinear systems operating along a specific trajectory. For an example, see LTV Model of Two-Link Robot.
Nonlinear systems operating near steady-state conditions that change in time.
Use ltvss
to construct LTV models whose dynamics are described by a
MATLAB® function (the data function). Use ssInterpolant
to
construct LTV models that interpolate LTI snapshots as a function of time. See LPV and LTV Models for functions and
operations applicable to ltvss
objects.
Creation
Syntax
Description
creates a continuous-time LTV model. ltvSys
= ltvss(DataFcn
)DataFcn
is the name of or a
handle to the data function, the user-defined MATLAB function that calculates the matrices and offsets for given
t or k values.
sets properties of the linear time-varying model using one or more name-value pair
arguments. Use this syntax with any of the previous input-argument combinations.ltvSys
= ltvss(___,Name=Value
)
Input Arguments
DataFcn
— Data function
function name | function handle
User-defined MATLAB function for calculating matrices and offsets, specified as a function name (character vector or string) or a function handle. The data function must be of the following form.
Continuous time —
Discrete time —
The Delay
argument is a
structure with fields Input
and Output
specifying the delays in input and output channels, respectively. (since R2024a)
For more information about defining a data function, see Data Function.
This input sets the value of property DataFunction
.
ts
— Sample time
scalar
Sample time, specified as a scalar. For more information, see the
Ts
property.
tcheck
— Time test value
scalar
Test value for time, specified as a scalar. The object evaluates
DataFcn
at tcheck
to check if it is valid
and to obtain the number of states, inputs, and outputs. By default,
ltvss
uses tcheck
= 0.
Properties
DataFunction
— Data function
[] (default) | function handle
Data function for calculating the model data, specified as a function handle. The data function must be of the following form.
Continuous time
[A,B,C,D,E,dx0,x0,u0,y0,Delay] = DataFcn(t)
Discrete time
[A,B,C,D,E,dx0,x0,u0,y0,Delay] = DataFcn(k)
Here, the input
k
is an integer index that counts the number of sampling periodsTs
. The absolute time is given byt
=k*Ts
.
To pass additional input arguments, use an anonymous function as follows.
DataFcn = @(t) myFunction(t,arg1,arg2,...)
You can set all output arguments except A
, B
,
C
, D
to []
when absent for
t
values.
For more information, see Data Function.
StateName
— State names
' '
(default) | character vector | cell array of character vectors
State names, specified as one of the following:
Character vector — For first-order models, for example,
'velocity'
.Cell array of character vectors — For models with two or more states.
StateName
is empty ' '
for all states by
default.
StatePath
— State path
' '
(default) | character vector | cell array of character vectors
State path to facilitate state block path management in linearization, specified as one of the following:
Character vector — For first-order models
Cell array of character vectors — For models with two or more states
StatePath
is empty ' '
for all states by
default.
StateUnit
— State units
' '
(default) | character vector | cell array of character vectors
State units, specified as one of the following:
Character vector — For first-order models, for example,
'm/s'
Cell array of character vectors — For models with two or more states
Use StateUnit
to keep track of the units of each state.
StateUnit
has no effect on system behavior.
StateUnit
is empty ' '
for all states by
default.
Ts
— Sample time
0
(default) | positive scalar | -1
Sample time, specified as:
0
for continuous-time systems.A positive scalar representing the sampling period of a discrete-time system. Specify
Ts
in the time unit specified by theTimeUnit
property.-1
for a discrete-time system with an unspecified sample time.
TimeUnit
— Time variable units
'seconds'
(default) | 'nanoseconds'
| 'microseconds'
| 'milliseconds'
| 'minutes'
| 'hours'
| 'days'
| 'weeks'
| 'months'
| 'years'
| ...
Time variable units, specified as one of the following:
'nanoseconds'
'microseconds'
'milliseconds'
'seconds'
'minutes'
'hours'
'days'
'weeks'
'months'
'years'
Changing TimeUnit
has no effect on other properties, but changes the overall system behavior. Use chgTimeUnit
to convert between time units without modifying system behavior.
InputName
— Input channel names
''
(default) | character vector | cell array of character vectors
Input channel names, specified as one of the following:
A character vector, for single-input models.
A cell array of character vectors, for multi-input models.
''
, no names specified, for any input channels.
Alternatively, you can assign input names for multi-input models using automatic vector
expansion. For example, if sys
is a two-input model, enter the
following.
sys.InputName = 'controls';
The input names automatically expand to {'controls(1)';'controls(2)'}
.
You can use the shorthand notation u
to refer to the InputName
property. For example, sys.u
is equivalent to sys.InputName
.
Use InputName
to:
Identify channels on model display and plots.
Extract subsystems of MIMO systems.
Specify connection points when interconnecting models.
InputUnit
— Input channel units
''
(default) | character vector | cell array of character vectors
Input channel units, specified as one of the following:
A character vector, for single-input models.
A cell array of character vectors, for multi-input models.
''
, no units specified, for any input channels.
Use InputUnit
to specify input signal units. InputUnit
has no effect on system behavior.
InputGroup
— Input channel groups
structure
Input channel groups, specified as a structure. Use InputGroup
to assign
the input channels of MIMO systems into groups and refer to each group by name. The
field names of InputGroup
are the group names and the field values
are the input channels of each group. For example, enter the following to create input
groups named controls
and noise
that include input
channels 1
and 2
, and 3
and
5
, respectively.
sys.InputGroup.controls = [1 2]; sys.InputGroup.noise = [3 5];
You can then extract the subsystem from the controls
inputs to all outputs
using the following.
sys(:,'controls')
By default, InputGroup
is a structure with no fields.
OutputName
— Output channel names
''
(default) | character vector | cell array of character vectors
Output channel names, specified as one of the following:
A character vector, for single-output models.
A cell array of character vectors, for multi-output models.
''
, no names specified, for any output channels.
Alternatively, you can assign output names for multi-output models using automatic vector
expansion. For example, if sys
is a two-output model, enter the
following.
sys.OutputName = 'measurements';
The output names automatically expand to {'measurements(1)';'measurements(2)'}
.
You can also use the shorthand notation y
to refer to the OutputName
property. For example, sys.y
is equivalent to sys.OutputName
.
Use OutputName
to:
Identify channels on model display and plots.
Extract subsystems of MIMO systems.
Specify connection points when interconnecting models.
OutputUnit
— Output channel units
''
(default) | character vector | cell array of character vectors
Output channel units, specified as one of the following:
A character vector, for single-output models.
A cell array of character vectors, for multi-output models.
''
, no units specified, for any output channels.
Use OutputUnit
to specify output signal units. OutputUnit
has no effect on system behavior.
OutputGroup
— Output channel groups
structure
Output channel groups, specified as a structure. Use OutputGroup
to
assign the output channels of MIMO systems into groups and refer to each group by name.
The field names of OutputGroup
are the group names and the field
values are the output channels of each group. For example, create output groups named
temperature
and measurement
that include
output channels 1
, and 3
and 5
,
respectively.
sys.OutputGroup.temperature = [1]; sys.OutputGroup.measurement = [3 5];
You can then extract the subsystem from all inputs to the measurement
outputs using the following.
sys('measurement',:)
By default, OutputGroup
is a structure with no fields.
Name
— System name
''
(default) | character vector
System name, specified as a character vector. For example, 'system_1'
.
Notes
— User-specified text
{}
(default) | character vector | cell array of character vectors
User-specified text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, 'System is MIMO'
.
UserData
— User-specified data
[]
(default) | any MATLAB data type
User-specified data that you want to associate with the system, specified as any MATLAB data type.
Object Functions
Sampling and Interpolation
sample | (Not recommended) Sample linear parameter-varying or time-varying dynamics |
ssInterpolant | Build gridded LTV or LPV model from state-space data |
Time Response Simulation
Model Interconnection
Examples
Continuous-Time Linear Time-Varying Model
Create a continuous-time SISO linear time-varying model.
This example uses a first-order model. The matrices and offsets are given by:
, , , ,
.
These matrices and offsets are defined in the ltvssDataFcn.m
data function provided with this example.
Create an LTV model.
ltvSys = ltvss(@ltvssDataFcn)
Continuous-time state-space LTV model with 1 outputs, 1 inputs, and 1 states.
View the data function.
type ltvssDataFcn.m
function [A,B,C,D,E,dx0,x0,u0,y0,Delays] = ltvssDataFcn(t) % SISO, first order A = -(1+0.5*sin(t)); B = 1; C = 1; D = 0; E = []; dx0 = []; x0 = []; u0 = []; y0 = 0.1*sin(5*t); Delays = [];
Discrete-Time Linear Time-Varying Model
Create a discrete-time linear time-varying model.
This example uses a model with the matrices and offsets defined as:
These matrices and offsets are defined in the ltvFcnDiscrete.m
data function provided with this example.
Specify the properties and create an LTV model.
Ts = 0.01; DataFcn = @ltvFcnDiscrete; ltvSys = ltvss(DataFcn,Ts)
Discrete-time state-space LTV model with 1 outputs, 1 inputs, and 1 states.
You can set additional properties of the model using dot notation
ltvSys.InputName = 'u'; ltvSys.OutputName = 'y';
View the data function.
type ltvFcnDiscrete.m
function [A,B,C,D,E,dx0,x0,u0,y0,Delays] = ltvFcnDiscrete(k) A = sin(0.1*k); B = 1; C = 1; D = 0; E = []; dx0 = []; x0 = []; u0 = []; y0 = 0.1*sin(k); Delays = [];
Time Response of Linear Time-Varying Models
For this example, ltvssDataFcn.m
defines the matrices and offsets of a MIMO system.
Create an LTV model.
ltvSys = ltvss(@ltvssDataFcn);
Simulate the response of this model to an arbitrary input from t = 0 to t = 10 seconds.
t = (0:.01:10)'; u = sin(0.2*t); x0 = 2;
Simulate and plot the response.
[y,~,x] = lsim(ltvSys,u,t,x0); plot(t,y)
Now, simulate the model to step and impulse responses.
Create an option set using RespConfig
to specify initial offset, and state values.
respOpt = RespConfig(InitialState=x0,Delay=1);
Compute the step response.
step(ltvSys,t,respOpt)
Compute the impulse response.
impulse(ltvSys,t,respOpt)
View the data function.
type ltvssDataFcn.m
function [A,B,C,D,E,dx0,x0,u0,y0,Delays] = ltvssDataFcn(t) % SISO, first order A = -(1+0.5*sin(t)); B = 1; C = 1; D = 0; E = []; dx0 = []; x0 = []; u0 = []; y0 = 0.1*sin(5*t); Delays = [];
Create LTV Model with Fixed and Varying Delays
Since R2024a
This example shows how to create a linear time-varying model containing fixed and varying input and output delays.
This example uses a model with the matrices, offsets, and delays defined in this data function.
function [A,B,C,D,E,dx0,x0,u0,y0,Delays] = ltvDataFcnDelay(t) A = [-2 1;1 -0.5*(2+sin(10*t))]; B = [1 cos(t);sin(5*t) 0]; C = [1,2*cos(3*t);-1 3;sqrt(t) 0]; D = [0 0;-1 1;0 0.5*sin(t)]; E = []; dx0 = [cos(10*t)/(1+0.1*t);0]; x0 = [0;sqrt(t)/(1+0.1*t)]; u0 = [sin(10*t);1]; y0 = [1/(1+t);-1;cos(3*t)]; Delays.Input = [NaN;abs(cos(0.3*t))]; Delays.Output = [1.7;NaN;abs(sin(0.5*t))]; end
The delays argument is a structure with fields Input
and Output
specifying the delays corresponding input and output channels, respectively. This data function defines the delays as follows:
No delay at all times in the first input channel and a varying delay of s in the second input channel.
Fixed delay of 1.7 s in the first output channel, no delay in the second output channel, and a varying delay of s in the third output channel.
Create the LTV model.
ltvsysD = ltvss(@ltvDataFcnDelay)
Continuous-time state-space LTV model with 3 outputs, 2 inputs, and 2 states. Model Properties
The software manages and propagates specified delays when you perform operations such as sampling.
Sample the model at two values of time.
t = [3,5]; sys = psample(ltvsysD,t)
sys(:,:,1,1) [Time=3] = A = x1 x2 x1 -2 1 x2 1 -0.506 B = u1 u2 x1 1 -0.99 x2 0.6503 0 C = x1 x2 y1 1 -1.822 y2 -1 3 y3 1.732 0 D = u1 u2 y1 0 0 y2 -1 1 y3 0 0.07056 Input delays (seconds): 0 0.622 Output delays (seconds): 1.7 0 0.997 sys(:,:,1,2) [Time=5] = A = x1 x2 x1 -2 1 x2 1 -0.8688 B = u1 u2 x1 1 0.2837 x2 -0.1324 0 C = x1 x2 y1 1 -1.519 y2 -1 3 y3 2.236 0 D = u1 u2 y1 0 0 y2 -1 1 y3 0 -0.4795 Input delays (seconds): 0 0.0707 Output delays (seconds): 1.7 0 0.598 1x2 array of continuous-time state-space models. Model Properties
The sampled LTV model contains fixed delays and delays varying with time as defined in the data function.
Version History
Introduced in R2023aR2024a: Data Function: Support for specifying input and output delays
Use the Delay
argument of the data function to specify fixed or
varying delays. The Delay
argument is a structure with fields
Input
and Output
specifying the delays in input and
output channels, respectively. Set Delay.Input
and
Delay.Output
to vectors with length equal to the number of input and
output channels, respectively. To indicate the absence of delay in a particular channel at
all times, set the corresponding value in the vector to NaN
. For example,
for a three-input system, Delay.Input = [0.1 NaN sin(0.2*t)]
defines a
fixed delay in the first input channel, no delay in the second channel, and a varying delay
in the third input channel. For more information about the data function, see Data Function
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