Identify Plant from Data at the Command Line

This example shows how to identify a plant model at the command line. For information on identifying models using the System Identification app, see Identify Linear Models Using System Identification App (System Identification Toolbox).

Load the measured input/output data.

load plantIO

This command imports the plant input signal, u, plant output signal, y, and sample time, Ts to the MATLAB® workspace.

Create an iddata object from the input and output data.

mydata = iddata(y,u,Ts);

You can optionally assign channel names and units for the input and output signals.

mydata.InputName = "Voltage";
mydata.InputUnit = "V";
mydata.OutputName = "Position";
mydata.OutputUnit = "cm";

Typically, you must preprocess identification I/O data before estimating a model. For this example, remove the offsets from the input and output signals by detrending the data.

mydatad = detrend(mydata);

You can also remove offsets by creating an ssestOptions object and specifying the InputOffset and OutputOffset options.

For this example, estimate a second-order, linear state-space model using the detrended data. To estimate a discrete-time model, specify the sample time as Ts.

ss1 = ssest(mydatad,2,Ts=Ts)

ss1 =
  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
   x1   0.8942  -0.1575
   x2   0.1961   0.7616
 
  B = 
         Voltage
   x1  6.008e-05
   x2   -0.01219
 
  C = 
                  x1       x2
   Position    38.24  -0.3835
 
  D = 
             Voltage
   Position        0
 
  K = 
       Position
   x1   0.03572
   x2    0.0223
 
Sample time: 0.1 seconds

Parameterization:
   FREE form (all coefficients in A, B, C free).
   Feedthrough: none
   Disturbance component: estimate
   Number of free coefficients: 10
   Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                             
Estimated using SSEST on time domain data "mydatad".
Fit to estimation data: 89.85% (prediction focus)   
FPE: 0.0156, MSE: 0.01541                           
 

You can use this identified plant as the internal prediction model for your MPC controller. When you do so, the controller converts the identified model to a discrete-time, state-space model.

By default, the MPC controller discards any unmeasured noise components from your identified model. To configure noise channels as unmeasured disturbances, you must first create an augmented state-space model from your identified model. For example:

ss2 = ss(ss1,"augmented")

ss2 =
 
  A = 
            x1       x2
   x1   0.8942  -0.1575
   x2   0.1961   0.7616
 
  B = 
          Voltage  v@Position
   x1   6.008e-05    0.004448
   x2    -0.01219    0.002777
 
  C = 
                  x1       x2
   Position    38.24  -0.3835
 
  D = 
                Voltage  v@Position
   Position           0      0.1245
 
Input groups:           
      Name      Channels
    Measured       1    
     Noise         2    
                        
Sample time: 0.1 seconds
Discrete-time state-space model.

This command creates a state-space model, ss2, with two input groups, Measured and Noise, for the measured and noise inputs respectively. When you import the augmented model into your MPC controller, channels in the Noise input group are defined as unmeasured disturbances.

Working with Impulse-Response Models

You can use System Identification Toolbox software to estimate finite step-response or finite impulse-response (FIR) plant models using measured data. Such models, also known as nonparametric models, are easy to determine from plant data ([1] and [2]) and have intuitive appeal.

Use the impulseest (System Identification Toolbox) function to estimate an FIR model from measured data. This function generates the FIR coefficients encapsulated as an idtf (System Identification Toolbox) object; that is, a transfer function model with only numerator coefficients. impulseest is especially effective in situations where the input signal used for identification has low excitation levels. To design a model predictive controller for this plant, you can convert the identified FIR plant model to a numeric LTI model. However, this conversion usually yields a high-order plant, which can degrade the controller design. For example, the numerical precision issues with high-order plants can affect estimator design. This result is particularly an issue for MIMO systems.

Model predictive controllers work best with low-order parametric models. Therefore, to design a model predictive controller using measured plant data, you can: