noise2meas

Noise component of model

Syntax

noise_model = noise2meas(sys)
noise_model = noise2meas(sys,noise)


Description

noise_model = noise2meas(sys) returns the noise component, noise_model, of a linear identified model, sys. Use noise2meas to convert a time-series model (no inputs) to an input/output model. The converted model can be used for linear analysis, including viewing pole/zero maps, and plotting the step response.

noise_model = noise2meas(sys,noise) specifies the noise variance normalization method.

Input Arguments

sys	Identified linear model.
noise	Noise variance normalization method, specified as one of the following values: 'innovations' — Noise sources are not normalized and remain as the innovations process. 'normalize' — Noise sources are normalized to be independent and of unit variance. Default: 'innovations'

Output Arguments

noise_model

Noise component of sys. sys represents the system $$y(t)=Gu(t)+He(t)$$ G is the transfer function between the measured input, u(t), and the output, y(t). H is the noise model and describes the effect of the disturbance, e(t), on the model’s response. An equivalent state-space representation of sys is $$\begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)+Ke(t)\\ y(t)=Cx(t)+Du(t)+e(t)\\ e(t)=Lv(t)\end{array}$$ v(t) is white noise with independent channels and unit variances. The white-noise signal e(t) represents the model’s innovations and has variance LLT. The noise-variance data is stored using the NoiseVariance property of sys. If noise is 'innovations', then noise2meas returns H and noise_model represents the system $$y(t)=He(t)$$ An equivalent state-space representation of noise_model is $$\begin{array}{l}\dot{x}(t)=Ax(t)+Ke(t)\\ y(t)=Cx(t)+e(t)\end{array}$$ noise2meas returns the noise channels of sys as the input channels of noise_model. The input channels are named using the format 'e@yk', where yk corresponds to the OutputName property of an output. The measured input channels of sys are discarded and the noise variance is set to zero. If noise is 'normalize', then noise2meas first normalizes $$e(t)=Lv(t)$$ noise_model represents the system $$y(t)=HLv(t)$$ or, equivalently, in state-space representation $$\begin{array}{l}\dot{x}(t)=Ax(t)+KLv(t)\\ y(t)=Cx(t)+Lv(t)\end{array}$$ The input channels are named using the format 'v@yk', where yk corresponds to the OutputName property of an output. The model type of noise_model depends on the model type of sys. noise_model is an idtf model if sys is an idproc model. noise_model is an idss model if sys is an idgrey model. noise_model is the same type of model as sys for all other model types. To obtain the model coefficients of noise_model in state-space form, use ssdata. Similarly, to obtain the model coefficients in transfer-function form, use tfdata.

Examples

Convert Noise Component of Linear Identified Model into Input/Output Model

Convert a time-series model to an input/output model that may be used by linear analysis tools.

Identify a time-series model.

load iddata9 z9
sys = ar(z9,4,'ls');

sys is an idpoly model with no inputs.

Convert sys to a measured model.

noise_model = noise2meas(sys);

noise_model is an idpoly model with one input.

You can use noise_model for linear analysis functions such as step, iopzmap, etc.

Normalizing Noise Variance

Convert an identified linear model to an input/output model, and normalize its noise variance.

Identify a linear model using data.

load twotankdata;
z = iddata(y,u,0.2);
sys = ssest(z,4);

sys is an idss model, with a noise variance of 6.6211e-06. The value of $$L$$ is sqrt(sys.NoiseVariance), which is 0.0026.

View the disturbance matrix.

sys.K

ans = 4×1

    0.2719
   -1.6570
   -0.6318
   -0.2877

Obtain a model that absorbs the noise variance of sys.

noise_model_normalize = noise2meas(sys,'normalize');

noise_model_normalize is an idpoly model.

View the $$B$$ matrix for noise_model_normalize

noise_model_normalize.B

ans = 4×1

    0.0007
   -0.0043
   -0.0016
   -0.0007

As expected, noise_model_normalize.B is equal to L*sys.K.

Compare the bode response with a model that ignores the noise variance of sys.

noise_model_innovation = noise2meas(sys,'innovations');
bodemag(noise_model_normalize,noise_model_innovation);
legend('Normalized noise variance','Ignored noise variance');

The difference between the bode magnitudes of the noise_model_innovation and noise_model_normalized is approximately 51 dB. As expected, the magnitude difference is approximately equal to 20*log10(L).

Version History

Introduced in R2012a