Generally, real systems are nonlinear. To design an MPC controller for a nonlinear system, you must 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. The Simulink environment provides two ways to accomplish this:
Note: Simulink Control Design™ software must be installed to linearize nonlinear Simulink models.
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,
. The nominal feed concentration is
, and the nominal feed temperature is 300 K. Create 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; op1 = findop('CSTR_OpenLoop',opspec);
Operating Point Search Report: --------------------------------- Operating Report for the Model CSTR_OpenLoop. (Time-Varying Components Evaluated at time t=0) Operating point specifications were successfully met. States: ---------- (1.) CSTR_OpenLoop/CSTR/C_A x: 2 dx: -4.6e-12 (0) (2.) CSTR_OpenLoop/CSTR/T_K x: 373 dx: 5.49e-11 (0) Inputs: ---------- (1.) CSTR_OpenLoop/Coolant Temperature u: 299 [-Inf Inf] Outputs: ---------- (1.) CSTR_OpenLoop/CSTR y: 2 (2)
The calculated operating point is
T_K = 373 K. Notice that the steady-state coolant temperature is also given as 299 K, which is the nominal value of the manipulated variable of the MPC controller.
Values of known inputs, use the
Input.u fields of
Initial guesses for state values, use the
State.x field of
For example, the following code specifies the coolant temperature as 305 K and initial guess values of the
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: --------------------------------- Operating Report for the Model CSTR_OpenLoop. (Time-Varying Components Evaluated at time t=0) Operating point specifications were successfully met. States: ---------- (1.) CSTR_OpenLoop/CSTR/C_A x: 1.78 dx: -8.88e-15 (0) (2.) CSTR_OpenLoop/CSTR/T_K x: 377 dx: 1.14e-13 (0) Inputs: ---------- (1.) CSTR_OpenLoop/Coolant Temperature u: 305 Outputs: None ----------
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/Feed Concentration', 1, 'input'); io(2) = linio('CSTR_OpenLoop/Feed Temperature', 1, 'input'); io(3) = linio('CSTR_OpenLoop/Coolant Temperature', 1, 'input'); io(4) = linio('CSTR_OpenLoop/CSTR', 1, 'output'); io(5) = linio('CSTR_OpenLoop/CSTR', 2, 'output');
Linearize the model using the specified operating point,
op1, and input/output signals,
sys = linearize('CSTR_OpenLoop', op1, io)
sys = a = C_A T_K C_A -5 -0.3427 T_K 47.68 2.785 b = Feed Concent Feed Tempera Coolant Temp C_A 1 0 0 T_K 0 1 0.3 c = C_A T_K CSTR/1 0 1 CSTR/2 1 0 d = Feed Concent Feed Tempera Coolant Temp CSTR/1 0 0 0 CSTR/2 0 0 0 Continuous-time state-space model.
This example shows how to linearize a Simulink model using the Linear Analysis Tool, provided by the Simulink Control Design product.
For this example, the CSTR model,
Open Simulink Model
sys = 'CSTR_OpenLoop'; open_system(sys)
Open Linear Analysis Tool
In the Simulink model window, select Analysis > Control Design > Linear Analysis.
Specify Linearization Inputs and Outputs
The linearization inputs and outputs are already specified for
The input signals correspond to the outputs from the
Feed Temperature, and
Temperature blocks. The output signals are the inputs to
CSTR Temperature and
Residual Concentration blocks.
To specify a signal as a:
Linearization input, right-click the signal in the Simulink model window and select Linear Analysis Points > Input Perturbation.
Linearization output, right-click the signal in the Simulink model window and select Linear Analysis Points > Output Measurement.
Specify Residual Concentration as Known Trim Constraint
In the Simulink model window, right-click the
signal from the
CSTR block. Select Linear Analysis Points > Trim
In the Linear Analysis Tool, in the Linear Analysis tab,
in the Operating Point drop-down list, select
In the Outputs tab:
Select the Known check box for
- 1 under CSTR_OpenLoop/CSTR.
Set the corresponding Value to
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.
op_trim1 to view the resulting
In the Edit dialog box, select the Input tab.
The coolant temperature at steady state is 299 K, as desired.
In the Linear Analysis tab, in the Operating
Point drop-down list, select
Click Step to linearize the model.
This creates the linear model
the Linear Analysis Workspace and generates a
step response for this model.
its operating point.
The step response from feed concentration to output
an interesting inverse response. An examination of the linear model
CSTR/2 is the residual CSTR concentration,
When the feed concentration increases,
initially because more reactant is entering, which increases the reaction
rate. This rate increase results in a higher reactor temperature
CSTR/1), which further increases the reaction
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, drag and drop it from the Linear Analysis Workspace to the MATLAB Workspace in the Linear Analysis Tool. You can now use your linear model to design an MPC controller.