Simulation and Prediction at the Command Line

Simulation and Prediction Commands

Note

If you estimated a linear model from detrended data and want to simulate or predict the output at the original operation conditions, use retrend to add trend data back into the simulated or predicted output.

Command	Description	Example
`compare`	Determine how closely the simulated model response matches the measured output signal. Plots simulated or predicted output of one or more models on top of the measured output. You should use an independent validation data set as input to the model.	To plot five-step-ahead predicted output of the model `mod` against the validation data `data`, use the following command: compare(data,mod,5) Note Omitting the third argument assumes an infinite horizon and results in the comparison of the simulated response to the input data.
`sim`	Simulate and plot the model output only.	To simulate the response of the model `model` using input data `data`, use the following command: sim(model,data)
`predict`	Predict and plot the model output only.	To perform one-step-ahead prediction of the response for the model `model` and input data `data`, use the following command: predict(model,data,1) Use the following syntax to compute `k`-step-ahead prediction of the output signal using model `m`: yhat = predict(m,[y u],k) `predict` computes the prediction results only over the time range of `data`. It does not forecast results beyond the available data range.
`forecast`	Forecast a time series into the future.	To forecast the value of a time series in an arbitrary number of steps into the future, use the following command: forecast(model,past_data,K) Here, `model` is a time series model, `past_data` is a record of the observed values of the time series, and `K` is the forecasting horizon.

Initial States in Simulation and Prediction

The process of computing simulated and predicted responses over a time range starts by using the initial conditions to compute the first few output values. sim, forecast, and predict commands provide defaults for handling initial conditions.

Simulation: Default initial conditions are zero for all model types except idnlgrey model, in which case the default initial conditions are the internal model initial states (model property x0). You can specify other initial conditions using the InitialCondition simulation option (see simOptions).

Use the compare command to validate models by simulation because its algorithm estimates the initial states of a model to optimize the model fit to a given data set.

If you use sim, the simulated and the measured responses might differ when the initial conditions of the estimated model and the system that measured the validation data set differ—especially at the beginning of the response. To minimize this difference, estimate the initial state values from the data using findstates and specify these initial states using the InitialCondition simulation option (see simOptions). For example, to compute the initial states that optimize the fit of the model m to the output data in z:

% Estimate the initial states
X0est = findstates(m,z);
% Simulate the response using estimated initial states
opt = simOptions('InitialCondition',X0est);
sim(m,z.InputData,opt)

Prediction: Default initial conditions depend on the type of model. You can specify other initial conditions using the InitialCondition option (see predictOptions). For example, to compute the initial states that optimize the 1-step-ahead predicted response of the model m to the output data z:

opt = predictOptions('InitialCondition','estimate');
[Yp,X0est] = predict(m,z,1,opt);

This command returns the estimated initial states as the output argument X0est. For information about other ways to specify initials states, see the predictOptions reference page.

Simulate a Continuous-Time State-Space Model

Open Live Script

This example shows how to simulate a continuous-time state-space model using a random binary input u and a sample time of 0.1 s.

Consider the following state-space model:

$\begin{array}{l} x_{}^{˙} = [\begin{array}{cccccccccccccccccccc} - 1 & 1 \\ - 0.5 & 0 \end{array}] x + [\begin{array}{cccccccccccccccccccc} 1 \\ 0.5 \end{array}] u + [\begin{array}{cccccccccccccccccccc} 0.5 \\ 0.5 \end{array}] e \\ y = [\begin{array}{cccccccccccccccccccc} 1 & 0 \end{array}] x + e \end{array}$

where e is Gaussian white noise with variance 7.

Create a continuous-time state-space model.

A = [-1 1; -0.5 0];
B = [1;0.5]; 
C = [1 0];
D = 0;
K = [0.5;0.5];
% Ts = 0 indicates continuous time
model_ss = idss(A,B,C,D,K,'Ts',0,'NoiseVariance',7);

Create a random binary input.

u = idinput(400,'rbs',[0 0.3]);

Create an iddata object with empty output to represent just the input signal.

data = iddata([],u);
data.ts = 0.1;

Simulate the output using the model

opt = simOptions('AddNoise',true); 
y = sim(model_ss,data,opt);

Simulate Model Output with Noise

Open Live Script

This example shows how you can create input data and a model, and then use the data and the model to simulate output data.

In this example, you create the following ARMAX model with Gaussian noise e:

$\begin{array}{l} y (t) - 1.5 y (t - 1) + 0.7 y (t - 2) = \\ u (t - 1) + 0.5 u (t - 2) + e (t) - e (t - 1) + 0.2 e (t - 1) \end{array}$

Then, you simulate output data with random binary input u.

Create an ARMAX model.

m_armax = idpoly([1 -1.5 0.7],[0 1 0.5],[1 -1 0.2]);

Create a random binary input.

u = idinput(400,'rbs',[0 0.3]);

Simulate the output data.

opt = simOptions('AddNoise',true);
y = sim(m_armax,u,opt);

The 'AddNoise' option specifies to include in the simulation the Gaussian noise e present in the model. Set this option to false (default behavior) to simulate the noise-free response to the input u , which is equivalent to setting e to zero.

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