This example shows how to specify a custom objective function for a model signal. You calculate the objective function value using a variable that models parameter uncertainty.

**Competitive Population Dynamics Model**

The Simulink model `sdoPopulation`

models a simple two-organism ecology using the competitive Lotka-Volterra equations:

is the population size of the n-th organism.

is the inherent per capita growth rate of each organism.

is the competitive delay for each organism.

is the carrying capacity of the organism environment.

is the proximity of the two populations and how strongly they affect each other.

The model uses normalized units.

Open the model.

```
open_system('sdoPopulation')
```

The two-dimensional signal, `P`

, models the population sizes for `P1`

(first element) and `P2`

(second element). The model is initially configured with one organism, `P1`

, dominating the ecology. The `Population`

scope shows the `P1`

population oscillating between high and low values, while `P2`

is constant at 0.1. The `Population Phase Portrait`

block shows the population sizes of the two organisms in relation to each other.

**Population Stabilization Design Problem**

Tune the , , and values to meet the following design requirements:

Minimize the population range, that is, the maximum difference between

`P1`

and`P2`

.

Stabilize

`P1`

and`P2`

, that is, ensure that neither organism population dies off or grows extremely large.

You must tune the parameters for different values of the carrying capacity, . This ensures robustness to environment carrying-capacity uncertainty.

**Open Response Optimization Tool**

Double-click the `Open Optimization Tool`

block in the model to open a pre configured Response Optimization tool session. The session specifies the following variables:

`DesignVars`

- Design variables set for the , and model parameters.

`K_unc`

- Uncertain parameter modeling the carrying capacity of the organism environment ( ).`K_unc`

specifies the nominal value and two sample values.

`P1`

and`P2`

- Logged signals representing the populations of the two organisms.

**Specify Custom Signal Objective Function**

Specify a custom requirement to minimize the maximum difference between the two population sizes. Apply this requirement to the `P1`

and `P2`

model signals.

Open the Create Requirement dialog box. In the

**New**list, select**Custom Requirement**.Specify the following in the Create Requirement dialog box:

**Name**- Enter`PopulationRange`

.**Type**- Select**Minimize the function output**from the list.**Function**- Enter`@sdoPopulation_PopRange`

. For more information about this function, see**Custom Signal Objective Function Details**.**Select Signals and Systems to Bound (Optional)**- Select the`P1`

and`P2`

check boxes.

3. Click **OK**.

A new variable, `PopulationRange`

, appears in the **Response Optimization Tool Workspace**.

**Custom Signal Objective Function Details**

`PopulationRange`

uses the `sdoPopulation_PopRange`

function. This function computes the maximum difference between the populations, across different environment carrying capacity values. By minimizing this value, you can achieve both design goals. The function is called by the optimizer at each iteration step.

To view the function, type `edit sdoPopulation_PopRange`

. The following discusses details of this function.

**Input/Output**

The function accepts `data`

, a structure with the following fields:

`DesignVars`

- Current iteration values of , and .

`Nominal`

- Logged signal data, obtained by simulating the model using parameter values specified by`data.DesignVars`

and nominal values for all other parameters. The`Nominal`

field is itself a structure with fields for each logged signal. The field names are the logged signal names. The custom requirement uses the logged signals,`P1`

and`P2`

. Therefore,`data.Nominal.P1`

and`data.Nominal.P2`

are timeseries objects corresponding to`P1`

and`P2`

.

`Uncertain`

- Logged signal data, obtained by simulating the model using the sample values of the uncertain variable`K_unc`

. The`Uncertain`

field is a vector of`N`

structures, where`N`

is the number of sample values specified for`K_unc`

. Each element of this vector is similar to`data.Nominal`

and contains simulation results obtained from a corresponding sample value specified for`K_unc`

.

The function returns the maximum difference between the population sizes across different carrying capacities. The following code snippet in the function performs this action:

val = max(maxP(1)-minP(2),maxP(2)-minP(1));

**Data Time Range**

When computing the design goals, discard the initial population growth data to eliminate biases from the initial-condition. The following code snippet in the function performs this action:

%Get the population data tMin = 5; %Ignore signal values prior to this time iTime = data.Nominal.P1.Time > tMin; sigData = [data.Nominal.P1.Data(iTime), data.Nominal.P2.Data(iTime)];

`iTime`

represents the time interval of interest, and the columns of `sigData`

contain `P1`

and `P2`

data for this interval.

**Optimization for Different Values of Carrying Capacity**

The function includes the effects of varying the carrying capacity by iterating through the elements of `data.Uncertain`

. The following code snippet in the function performs this action:

... for ct=1:numel(data.Uncertain) iTime = data.Uncertain(ct).P1.Time > tMin; sigData = [data.Uncertain(ct).P1.Data(iTime), data.Uncertain(ct).P2.Data(iTime)];

maxP = max([maxP; max(sigData)]); %Update maximum if new signals are bigger minP = min([minP; min(sigData)]); %Update minimum if new signals are smaller end ...

The maximum and minimal populations are obtained across all the simulations contained in `data.Uncertain`

.

**Optimize Design**

Click **Optimize**.

The optimization converges after a number of iterations.

The `P1,P2`

plot shows the population dynamics, with the first organism population in blue and the second organism population in red. The dotted lines indicate the population dynamics for different environment capacity values. The `PopulationRange`

plot shows that the maximum difference between the two organism populations reduces over time.

The `Population Phase Portrait`

block shows the populations initially varying, but they eventually converge to stable population sizes.

% Close the model bdclose('sdoPopulation')

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