# How do I estimate or optimize the parameters of my ODE system in MATLAB 8.1 (R2013a)?

I have a system of ordinary differential equations (ODE) with some unknown parameters (coefficients). I want to simultaneously solve the system of differential equations as well as optimize for the unknown parameters by minimizing an objective function that depends on the solution of the ODE system.

What is the best way to do this in MATLAB?

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There are two approaches that can be taken:

1) The optimization toolbox in MATLAB provides function such as FMINSEARCH, LSQNONLIN, FMINCON etc., which can be used for optimizing parameters while minimizing an objective function. In this case, the objective function will have to call another sub-routine which solves the differential equations using ODE solvers such as ODE23, ODE45, ODE23s, ODE113, or ODE15s. The ODE solver in turn will call the function where the differential equations are implemented.

Attached is a simple example that optimizes a single parameter for a 1 state simulation (that is a first order ODE). Note that the ‘optimization toolbox’ is needed to execute the example.

The workflow can be described as follows:

a) Choose an ODE solver (i.e. ode45 or ode15s), then write the update function for the differential equations.

b) Write an objective function that takes in the values of the parameters, solves the ODE for those particular values, and then calculates the cost function (such as the difference between the experimental and simulated data) that needs to be minimized.

c) Use an optimization function like LSQNONLIN or FMINCON to minimize the objective function.

d) Use this section of the documentation to address potential issues that go along with a numerical optimization routine wrapped around a numerical simulation.

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