# 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.

$$\begin{array}{c}E(t)\dot{x}={\delta}_{0}(t)+A(t)(x-{x}_{0}(t))+B(t)(u-{u}_{0}(t))\\ y(t)={y}_{0}(t)+C(t)(x-{x}_{0}(t))+D(t)(u-{u}_{0}(t))\end{array}$$

Here, *A*(*t*),
*B*(*t*), *C*(*t*),
*D*(*t*), and *E*(*t*)
are time-varying state-space matrices and
*δ*_{0}(*t*),
*x*_{0}(*t*),
*u*_{0}(*t*), and
*y*_{0}(*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.

$$\begin{array}{c}{E}_{k}{x}_{k+1}={\delta}_{0k}+{A}_{k}({x}_{k}-{x}_{0}{}_{k})+{B}_{k}({u}_{k}-{u}_{0}{}_{k})\\ {y}_{k}={y}_{0k}+{C}_{k}({x}_{k}-{x}_{0}{}_{k})+{D}_{k}({u}_{k}-{u}_{0}{}_{k})\end{array}$$

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 — $$[A,B,C,D,E,{\delta}_{0},{x}_{0},{u}_{0},{y}_{0},Delays]=f(t)$$

Discrete time — $$[A,B,C,D,E,{\delta}_{0},{x}_{0},{u}_{0},{y}_{0},Delays]=f(k)$$

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 periods`Ts`

. The absolute time is given by`t`

=`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 the`TimeUnit`

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:

$\mathit{A}=-\left(1+0.5\mathrm{sin}\left(\mathit{t}\right)\right)$, $\mathit{B}=1$, $\mathit{C}=1$, $\mathit{D}=0$,

${\mathit{y}}_{0}=0.1\mathrm{sin}\left(5\mathit{t}\right)$.

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:

$$\mathit{A}=\mathrm{sin}\left(0.1\mathit{k}\right),\mathit{B}=1,\mathit{C}=1,\mathit{D}=0$$

$${\mathit{y}}_{0}=0.1\mathrm{sin}\left(\mathit{k}\right)$$

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 $\mathrm{cos}\left(0.3\mathit{t}\right)$ 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 $\mathrm{sin}\left(0.5\mathit{t}\right)$ 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 R2023a**

### R2024a: 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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