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Linearize Simulink Models

Generally, real systems are nonlinear. To design an MPC controller for a nonlinear system, you can model the plant in Simulink®.

Although an MPC controller can regulate a nonlinear plant, the model used within the controller must be linear. In other words, the controller employs a linear approximation of the nonlinear plant. The accuracy of this approximation significantly affects controller performance.

To obtain such a linear approximation, you linearize the nonlinear plant at a specified operating point.

Note

Simulink Control Design™ software must be installed to linearize nonlinear Simulink models.

You can linearize a Simulink model:

Linearization Using MATLAB Code

This example shows how to obtain a linear model of a plant using a MATLAB script.

For this example the CSTR model, CSTR_OpenLoop, is linearized. The model inputs are the coolant temperature (manipulated variable of the MPC controller), limiting reactant concentration in the feed stream, and feed temperature. The model states are the temperature and concentration of the limiting reactant in the product stream. Both states are measured and used for feedback control.

Obtain Steady-State Operating Point

The operating point defines the nominal conditions at which you linearize a model. It is usually a steady-state condition.

Suppose that you plan to operate the CSTR with the output concentration, C_A, at 2kmol/m3. The nominal feed concentration is 10kmol/m3, and the nominal feed temperature is 300 K.

Create and visualize an operating point specification object to define the steady-state conditions.

opspec = operspec('CSTR_OpenLoop');
opspec = addoutputspec(opspec,'CSTR_OpenLoop/CSTR',2);
opspec.Outputs(1).Known = true;
opspec.Outputs(1).y = 2;
opspec
opspec = 
 Operating point specification for the Model CSTR_OpenLoop.
 (Time-Varying Components Evaluated at time t=0)

States: 
----------
      x       Known    SteadyState    Min    Max    dxMin    dxMax
    ______    _____    ___________    ___    ___    _____    _____

(1.) CSTR_OpenLoop/CSTR/C_A
    8.5695    false       true         0     Inf    -Inf      Inf 
(2.) CSTR_OpenLoop/CSTR/T_K
    311.27    false       true         0     Inf    -Inf      Inf 

Inputs: 
----------
    u    Known    Min     Max
    _    _____    ____    ___

(1.) CSTR_OpenLoop/Coolant Temperature
    0    false    -Inf    Inf

Outputs: 
----------
    y    Known    Min     Max
    _    _____    ____    ___

(1.) CSTR_OpenLoop/CSTR
    2    true     -Inf    Inf

Search for an operating point that satisfies the specifications.

op1 = findop('CSTR_OpenLoop',opspec);
 Operating point search report:
---------------------------------
opreport = 
 Operating point search report for the Model CSTR_OpenLoop.
 (Time-Varying Components Evaluated at time t=0)

Operating point specifications were successfully met.
States: 
----------
    Min      x       Max    dxMin        dx         dxMax
    ___    ______    ___    _____    ___________    _____

(1.) CSTR_OpenLoop/CSTR/C_A
     0          2    Inf      0      -4.5972e-12      0  
(2.) CSTR_OpenLoop/CSTR/T_K
     0     373.13    Inf      0        5.484e-11      0  

Inputs: 
----------
    Min       u       Max
    ____    ______    ___

(1.) CSTR_OpenLoop/Coolant Temperature
    -Inf    299.03    Inf

Outputs: 
----------
    Min    y    Max
    ___    _    ___

(1.) CSTR_OpenLoop/CSTR
     2     2     2 

The calculated operating point is C_A = 2kmol/m3 and T_K = 373 K. Notice that the steady-state coolant temperature is also given as 299 K, which is the nominal value of the input used to control the plant.

To specify:

  • Values of known inputs, use the Input.Known and Input.u fields of opspec

  • Initial guesses for state values, use the State.x field of opspec

For example, the following code specifies the coolant temperature as 305 K and initial guess values of the C_A and T_K states before calculating the steady-state operating point:

opspec = operspec('CSTR_OpenLoop');
opspec.States(1).x = 1;
opspec.States(2).x = 400;
opspec.Inputs(1).Known = true;
opspec.Inputs(1).u = 305;

op2 = findop('CSTR_OpenLoop',opspec)
 Operating point search report:
---------------------------------
opreport = 
 Operating point search report for the Model CSTR_OpenLoop.
 (Time-Varying Components Evaluated at time t=0)

Operating point specifications were successfully met.
States: 
----------
    Min      x       Max    dxMin        dx         dxMax
    ___    ______    ___    _____    ___________    _____

(1.) CSTR_OpenLoop/CSTR/C_A
     0     1.7787    Inf      0      -4.7962e-14      0  
(2.) CSTR_OpenLoop/CSTR/T_K
     0     376.54    Inf      0       5.4001e-13      0  

Inputs: 
----------
    Min     u     Max
    ___    ___    ___

(1.) CSTR_OpenLoop/Coolant Temperature
    305    305    305

Outputs: None 
----------
op2 = 
 Operating point for the Model CSTR_OpenLoop.
 (Time-Varying Components Evaluated at time t=0)

States: 
----------
      x   
    ______

(1.) CSTR_OpenLoop/CSTR/C_A
    1.7787
(2.) CSTR_OpenLoop/CSTR/T_K
    376.54

Inputs: 
----------
     u 
    ___

(1.) CSTR_OpenLoop/Coolant Temperature
    305

Specify Linearization Inputs and Outputs

If the linearization input and output signals are already defined in the model, as in CSTR_OpenLoop, then use the following to obtain the signal set.

io = getlinio('CSTR_OpenLoop');

Otherwise, specify the input and output signals as shown here.

io(1) = linio('CSTR_OpenLoop/Coolant Temperature',1,'input');
io(2) = linio('CSTR_OpenLoop/Feed Concentration',1,'input');
io(3) = linio('CSTR_OpenLoop/Feed Temperature',1,'input');
io(4) = linio('CSTR_OpenLoop/CSTR',1,'output');
io(5) = linio('CSTR_OpenLoop/CSTR',2,'output');

Linearize Model

Linearize the model using the specified operating point, op1, and input/output signals, io.

sys = linearize('CSTR_OpenLoop',op1,io)
sys =
 
  A = 
            C_A      T_K
   C_A       -5  -0.3427
   T_K    47.68    2.785
 
  B = 
        Coolant Temp  Feed Concent  Feed Tempera
   C_A             0             1             0
   T_K           0.3             0             1
 
  C = 
           C_A  T_K
   CSTR/1    0    1
   CSTR/2    1    0
 
  D = 
           Coolant Temp  Feed Concent  Feed Tempera
   CSTR/1             0             0             0
   CSTR/2             0             0             0
 
Continuous-time state-space model.

Linearize the model also around the operating point, op2, using the same input/output signals.

sys = linearize('CSTR_OpenLoop',op2,io)
sys =
 
  A = 
            C_A      T_K
   C_A   -5.622  -0.3458
   T_K     55.1    2.822
 
  B = 
        Coolant Temp  Feed Concent  Feed Tempera
   C_A             0             1             0
   T_K           0.3             0             1
 
  C = 
           C_A  T_K
   CSTR/1    0    1
   CSTR/2    1    0
 
  D = 
           Coolant Temp  Feed Concent  Feed Tempera
   CSTR/1             0             0             0
   CSTR/2             0             0             0
 
Continuous-time state-space model.

Linearization Using Model Linearizer in Simulink Control Design

This example shows how to linearize a Simulink model using the Model Linearizer, provided by the Simulink Control Design software.

Open Simulink Model

This example uses the CSTR model, CSTR_OpenLoop.

open_system('CSTR_OpenLoop')

Specify Linearization Inputs and Outputs

The linearization inputs and outputs are already specified for CSTR_OpenLoop. The input signals correspond to the outputs from the Feed Concentration, Feed Temperature, and Coolant Temperature blocks. The output signals are the inputs to the CSTR Temperature and Residual Concentration blocks.

To specify a signal as a linearization input or output, first open the Linearization tab. To do so, in the Simulink Apps gallery, click Linearization Manager. Then, in the Simulink model window, click the signal.

To specify the signal as a:

  • Linearization input, on the Linearization tab, in the Insert Analysis Points gallery, click Input Perturbation.

  • Linearization output, on the Linearization tab, in the Insert Analysis Points gallery, click Output Measurement.

Open Model Linearizer

To open the Model Linearizer, in the Apps gallery, click Model Linearizer.

Specify Residual Concentration as Known Trim Constraint

In the Simulink model window, click the CA output signal from the CSTR block. Then, on the Linearization tab, in the Insert Analysis Points gallery, click Trim Output Constraint.

In the Model Linearizer, on the Linear Analysis tab, select Operating Point > Trim Model.

In the Trim the model dialog box, on the Outputs tab:

  • Select the Known check box for Channel - 1 under CSTR_OpenLoop/CSTR.

  • Set the corresponding Value to 2 kmol/m3.

Create and Verify Operating Point

In the Trim the model dialog box, click Start trimming.

The operating point op_trim1 displays in the Linear Analysis Workspace.

Double click op_trim1 to view the resulting operating point.

In the Edit dialog box, select the Input tab.

The coolant temperature at steady state is 299 K, as desired.

Linearize Model

On the Linear Analysis tab, in the Operating Point drop-down list, select op_trim1.

Click Step to linearize the model.

This option creates the linear model linsys1 in the Linear Analysis Workspace and generates a step response for this model. linsys1 uses optrim1 as its operating point.

The step response from feed concentration to output CSTR/2 displays an interesting inverse response. An examination of the linear model shows that CSTR/2 is the residual CSTR concentration, C_A. When the feed concentration increases, C_A increases initially because more reactant is entering, which increases the reaction rate. This rate increase results in a higher reactor temperature (output CSTR/1), which further increases the reaction rate and C_A decreases dramatically.

Export Linearization Result

If necessary, you can repeat any of these steps to improve your model performance. Once you are satisfied with your linearization result, in the Model Linearizer, drag and drop it from the Linear Analysis Workspace to the MATLAB Workspace. You can now use your linear model to design an MPC controller.

See Also

(Simulink Control Design) | (Simulink Control Design)

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