Simulink is a toolbox of MATLAB that can be used for modeling, analyzing and simulating dynamical systems. Here, we focus on the use of Simulink on the use with ordinary differential equations.

Simulating the dynamical behavior of a system is important in engineering. Many systems can be written as a differential equation, such as:

$\displaystyle m\ddot{x}(t)+b\dot{x}(t)+kx(t)=u(t),$ (7.1)

describing the movement of a single mass, on which an actuation force $ u$ is applied. For some simple differential equations, trajectories $ x(t)$ of the system can be computed analytically. Many of these analytical solutions have been implemented in MATLAB and can be obtained with the command dsolve.

However, many systems found in engineering or physics are described by more complex differential equations, for which no analytical solution is known or even exists. Trajectories of these systems can be approximated numerically. Hereto, solvers are developed that approximate the change of the state variables over a small time step. Herewith, step by step, an approximation of the trajectory is found. For example, the MATLAB functions ode23 and ode45 are numerical solvers. Simulink is an other tool in MATLAB using numerical solvers.

Use of simulink has some advantages over the use of the functions ode23 and ode45. Simulink has a graphical interface, such that the structure of the system can be clearly visible. Inputs, such as a discontinuous signal $ u(t)$ for system (7.1) can be easily applied. Furthermore, real time applications are possible. Examples of real time applications are a control scheme to control a printer head motion, or a control scheme to control the flow of liquid in a heart simulator.

Before we start using Simulink, we first show how to represent a system such as (7.1) in a block diagram.

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Esteur 2010-03-22