Use Matrix-Based Data for Time-Domain System Identification

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The simplest way to represent data in System Identification Toolbox™ is to use numeric matrices, one matrix for input data, and one matrix for output data. The observations are along the rows and the channels are along the columns. This data format is often sufficient for identifying discrete-time models. Note that this format does not provide any information about the time vector. Hence this format is not recommended for identifying continuous-time models.

Estimate Discrete-Time Model

As a simple example, estimate a discrete-time model from matrix data. Specify the matrices as a comma-separated pair $u, y$ .

load sdata1 umat1 ymat1
sysd = n4sid(umat1, ymat1, 4)

sysd =
  Discrete-time identified state-space model:
    x(t+Ts) = A x(t) + B u(t) + K e(t)
       y(t) = C x(t) + D u(t) + e(t)
 
  A = 
              x1         x2         x3         x4
   x1     0.8392    -0.3129    0.02105    0.03743
   x2     0.4768     0.6671     0.1428   0.003757
   x3   -0.01951    0.08374   -0.09761      1.046
   x4  -0.003885   -0.02914    -0.8796   -0.03171
 
  B = 
              u1
   x1    0.02635
   x2   -0.03301
   x3  7.256e-05
   x4  0.0005861
 
  C = 
            x1       x2       x3       x4
   y1    69.08    26.64   -2.237  -0.5601
 
  D = 
       u1
   y1   0
 
  K = 
              y1
   x1   0.003282
   x2   0.009339
   x3  -0.003232
   x4   0.003809
 
Sample time: 1 seconds

Parameterization:
   FREE form (all coefficients in A, B, C free).
   Feedthrough: none
   Disturbance component: estimate
   Number of free coefficients: 28
   Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                           
Estimated using N4SID on time domain data "umat1".
Fit to estimation data: 76.33% (prediction focus) 
FPE: 1.21, MSE: 1.087

View the sample time.

sysd.Ts

ans = 1

Since the data does not provide the sample time, the software assumes that the data sample time is 1 second. Consequently, the model sample time, sysd.Ts, is also 1 second (Note that n4sid estimates a discrete-time model by default.)

Since the data does not provide the sample time information, you can specify different sample time for the model by using the 'Ts'/<value> name-value argument. Specify a model sample time of 0.1 seconds.

sysd = n4sid(umat1, ymat1, 4, 'Ts', 0.1);

The software assumes that the data sample time is also 0.1 seconds.

Estimate Continuous-Time Model

Simply specifying the sample time does not work if the goal is to specify a continuous-time model by setting Ts to 0.

sysc = n4sid(umat1, ymat1, 4, 'Ts', 0);

Warning: Data sample time is assumed to be 1 seconds. To specify a different value for the data sample time consider providing data using a timetable or an iddata object.

In this case, there is no direct way of specifying the data sample time. The software assumes that the data sample time is 1 second and issues a warning.

To estimate a continuous-time model with this data, you must convert it into a timetable or an iddata object so that you can specify the data sample time information.

Suppose that the true sample time of the data above is 0.1 hours. Convert the data to a timetable using the data sample time information.

N = size(umat1,1);
% (assume a start time of 1 hr)
t = hours(0.1*(1:N)');
dataTT = timetable(umat1,ymat1,'RowTimes',t);

You can now estimate the continuous model

sysc = n4sid(dataTT, 4, 'Ts', 0);
sysc.Ts

ans = 0

sysc.TimeUnit

ans = 
'hours'

Alternatively, instead of converting the data to a timetable, you can convert it to an iddata object. Note that the conversion command for iddata specifies output first, while the timetable command specifies input first.

dataIDDATA = iddata(ymat1, umat1, 0.1, 'TimeUnit', 'hours');
sysc = n4sid(dataIDDATA, 4, 'Ts', 0);
sysc.Ts

ans = 0

sysc.TimeUnit

ans = 
'hours'

Estimate Time Series Model

For time series data, that is, data containing only output data and no input data, specify [], $y$ .

systs = n4sid([],ymat1,4)

systs =
  Discrete-time identified state-space model:
    x(t+Ts) = A x(t) + K e(t)
       y(t) = C x(t) + e(t)
 
  A = 
             x1        x2        x3        x4
   x1    0.8293   -0.4941   -0.0201    0.2288
   x2    0.2122    0.6994  -0.03311    0.5703
   x3  -0.01299   0.03653   -0.8259    0.7642
   x4  0.007749  -0.04922   -0.3883   -0.1022
 
  C = 
            x1       x2       x3       x4
   y1   -74.48     14.6   0.6196  -0.1086
 
  K = 
               y1
   x1    -0.01566
   x2    0.006688
   x3  -0.0009932
   x4   0.0005739
 
Sample time: 1 seconds

Parameterization:
   FREE form (all coefficients in A, B, C free).
   Disturbance component: estimate
   Number of free coefficients: 24
   Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                          
Estimated using N4SID on time domain data.       
Fit to estimation data: 59.64% (prediction focus)
FPE: 3.425, MSE: 3.161

systs is a time series model that contains no $B$ or $D$ component.