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Compute Steady-State Operating Point from State Specifications

You can compute a steady-state operating point for a Simulink® model by specifying constraints on the model states, and finding a model operating condition that satisfies these constraints. For more information on steady-state operating points, see About Operating Points and Compute Steady-State Operating Points.

You can trim your model to meet state specifications interactively using the Linear Analysis Tool or programmatically at the MATLAB® command line. For each state in your model, you can specify a known value or you can constrain the state value using minimum and maximum bounds. If a state is not known, you can specify an initial guess. You can also specify which states must be at steady-state at the trimmed operating point.

You can also constrain the derivatives of states that are not at steady-state.

Compute Operating Point from State Specifications Using Linear Analysis Tool

This example shows how to compute a steady-state operating point by specifying known state values and constraints using the Linear Analysis Tool.

Open the Simulink model.

sys = 'scdspeed';
open_system(sys)

To open the Linear Analysis Tool, in the Simulink model window, select Analysis > Control Design > Linear Analysis.

In the Linear Analysis Tool, on the Linear Analysis tab, in the Operating Point drop-down list, select Trim Model.

By default, on the States tab, the software specifies both model states to be at equilibrium, as shown by the check marks in the Steady State column. both states are also specified as unknown values; that is, their steady-state values are calculated during trimming, with an initial guess specified in the Value column.

Change the second state, the engine angular velocity, to be a known value. In the Known column, select the corresponding row and, in the Value column, set the value to 180.

You can also specify bounds for model states during trimming. For this example, constrain the first state to be between 0.5 and 0.7. To do so, enter these values in the Minimum and Maximum columns, respectively.

To compute the operating point that meets these specifications, click Start trimming.

The software uses optimization to find the operating point that meets your specifications.

The Trim progress viewer shows the optimization progress and that the optimization algorithm terminated successfully. The (Maximum Error) column shows the maximum constraint violation at each iteration. The Block column shows the block to which the constraint violation applies.

The trimmed operating point, op_trim1, appears in the Linear Analysis Workspace.

To evaluate whether the resulting operating point values meet the specifications, in the Linear Analysis Workspace, double-click op_trim1.

In the Edit dialog box, on the State tab, the Actual Value for the first state falls within the Desired Value bounds, and the actual angular velocity is 180, as specified.

The Actual dx column shows the rates of change of the state values at the operating point. Since these values are at or near zero the states are not changing, showing that the operating point is in a steady state.

You can also constrain the derivatives of states that are not at steady state. Using such constraints, you can trim derivatives to known nonzero values or specify derivative tolerances for states that cannot reach steady state.

For example, suppose that you want to find the operating condition at which the engine angular velocity is 180 rad/s and the angular acceleration is 50 rad/s2. To do so, first open the Trim the model dialog box. In the Linear Analysis Tool, in the Operating Point drop-down list, select Trim Model.

In the Steady State column, clear the selection in the corresponding row. Then, in the dx Minimum and dx Maximum columns, set both state derivative bounds to 50.

To compute the operating point, click Start trimming.

In the Linear Analysis Tool, in the Linear Analysis Workspace, double-click op_trim2.

In the Edit dialog box, in the second row, the Actual dx column matches the Desired dx column. Therefore the operating point meets the specified state derivative constraints.

Compute Operating Point from State Specifications at Command Line

This example shows how to compute a steady-state operating point by specifying known state values and constraints.

Open the Simulink model.

mdl = 'scdspeed';
open_system(mdl)

Create a default operating point specification for the model.

opspec = operspec(mdl)
 Operating point specification for the Model scdspeed.
 (Time-Varying Components Evaluated at time t=0)

States: 
----------
(1.) scdspeed/Throttle & Manifold/Intake Manifold/p0 = 0.543 bar
	 spec:  dx = 0,  initial guess: 0.543
(2.) scdspeed/Vehicle Dynamics/w = T//J w0 = 209 rad//s
	 spec:  dx = 0,  initial guess: 209

Inputs: 
----------
(1.) scdspeed/Throttle  perturbation
	 initial guess: 0            

Outputs: None 
----------


By default, both states are specified to be at equilibrium, as shown by the dx = 0 specification. Both states are also specified as unknown values; that is, their steady-state values are calculated during trimming, with an initial guess defined in the specification.

Change the second state, the engine angular velocity, to be a known value, and view the updated state specifications.

opspec.States(2).Known = 1;
opspec.States
(1.) scdspeed/Throttle & Manifold/Intake Manifold/p0 = 0.543 bar
	 spec:  dx = 0,  initial guess: 0.543
(2.) scdspeed/Vehicle Dynamics/w = T//J w0 = 209 rad//s
	 spec:  dx = 0,  x:           209

The value defined in the second state specification is now the known state value and not an initial guess.

Find an operating point that meets these specifications.

op1 = findop(mdl,opspec);
 Operating point search report:
---------------------------------

 Operating point search report for the Model scdspeed.
 (Time-Varying Components Evaluated at time t=0)

Operating point specifications were successfully met.
States: 
----------
(1.) scdspeed/Throttle & Manifold/Intake Manifold/p0 = 0.543 bar
      x:         0.544      dx:      2.03e-13 (0)
(2.) scdspeed/Vehicle Dynamics/w = T//J w0 = 209 rad//s
      x:           209      dx:     -4.57e-13 (0)

Inputs: 
----------
(1.) scdspeed/Throttle  perturbation
      u:       0.00501    [-Inf Inf]

Outputs: None 
----------

The operating point search report shows that the specifications were met successfully, and that both states are at steady state as expected (dx = 0).

You can also specify bounds for model states during trimming. For example, modify the operating point specifications to trim the second state to a known value of 180, while constraining the first state to be between 0.5 and 0.7.

opspec.States(2).x = 180;
opspec.States(1).Min = 0.5;
opspec.States(1).Max = 0.7;

Find the operating point that meets these specifications.

op2 = findop(mdl,opspec);
 Operating point search report:
---------------------------------

 Operating point search report for the Model scdspeed.
 (Time-Varying Components Evaluated at time t=0)

Operating point specifications were successfully met.
States: 
----------
(1.) scdspeed/Throttle & Manifold/Intake Manifold/p0 = 0.543 bar
      x:          0.57      dx:      2.47e-10 (0)
(2.) scdspeed/Vehicle Dynamics/w = T//J w0 = 209 rad//s
      x:           180      dx:      2.03e-13 (0)

Inputs: 
----------
(1.) scdspeed/Throttle  perturbation
      u:        -0.881    [-Inf Inf]

Outputs: None 
----------

The operating point search report shows that the specifications were met successfully, and that the first state is within the specified bounds.

Finally, you can also constrain the derivatives of states that are not at steady state. Using such constraints, you can trim derivatives to known nonzero values or specify derivative tolerances for states that cannot reach steady state.

For example, suppose that you want to find the operating condition at which the engine angular velocity is 180 rad/s and the angular acceleration is 50 rad/s^2. To do so, disable the SteadyState specification for that state, and set both state derivative bounds to the same target value.

opspec.States(2).SteadyState = 0;
opspec.States(2).dxMin = 50;
opspec.States(2).dxMax = 50;

The updated state specifications show the new state derivative constraints.

opspec.States
(1.) scdspeed/Throttle & Manifold/Intake Manifold/p0 = 0.543 bar
	 spec:  dx = 0,  initial guess: 0.543
(2.) scdspeed/Vehicle Dynamics/w = T//J w0 = 209 rad//s
	 spec:  50 <= dx <= 50,  x:           180

Find an operating point that meets these updated specifications.

op3 = findop(mdl,opspec);
 Operating point search report:
---------------------------------

 Operating point search report for the Model scdspeed.
 (Time-Varying Components Evaluated at time t=0)

Operating point specifications were successfully met.
States: 
----------
(1.) scdspeed/Throttle & Manifold/Intake Manifold/p0 = 0.543 bar
      x:         0.666      dx:      2.01e-08 (0)
(2.) scdspeed/Vehicle Dynamics/w = T//J w0 = 209 rad//s
      x:           180      dx:            50 [50, 50]

Inputs: 
----------
(1.) scdspeed/Throttle  perturbation
      u:         0.163    [-Inf Inf]

Outputs: None 
----------

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