Accelerating the pace of engineering and science

# Documentation

## Specify a Time-Invariant State-Space Model

This example shows how to specify a time-invariant state-space model by passing a parameter-mapping function describing the model to ssm (i.e., implicitly specify a state-space model). The state model is AR(1), and there is no observation error. Set the initial state mean and variance, and specify that the state is stationary.

Create the function timeInvariantParamMap.m, which specifies how the parameters in params map to the state-space model matrices, the initial state values, and the type of state. Symbolically, the model is

$\begin{array}{c}{x}_{t}=\varphi {x}_{t-1}+\sigma {u}_{t}\\ {y}_{t}=a{x}_{t}\end{array}.$

```function [A,B,C,D,Mean0,Cov0,StateType] = timeInvariantParamMap(params)
% Time-invariant state-space model parameter mapping function example. This
% function maps the vector params to the state-space matrices (A, B, C, and
% D), the initial state value and the initial state variance (Mean0 and
% Cov0), and the type of state (StateType). The state model is AR(1)
% without observation error.
varu1 = exp(params(2)); % Positive variance constraint
A = params(1);
B = sqrt(vare1);
C = params(3);
D = [];
Mean0 = 0.5;
Cov0 = 100;
StateType = 0;
end

```

Specify the state-space model by passing the function timeInvariantParamMap as a function handle to ssm.

`Mdl = ssm(@timeInvariantParamMap)`
```Mdl =

The state space model is implicitly defined by the function:
@timeInvariantParamMap```

The software implicitly defines the state-space model. Usually, you cannot verify state-space models that you implicitly define.

Pass Mdl and response data to estimate to estimate the model parameters.