Systems and Control Theory
Master Degree Course in ELECTRONICS ENGINEERING
http://www.dii.unimore.it/~lbiagiotti/SystemsControlTheory.html
Introduction to SIMULINK
Luigi Biagiotti
e-mail: luigi.biagiotti@unimore.it
http://www.dii.unimore.it/~lbiagiotti
Simulink introduction
• Simulink (Simulation and Link) is an extension of MATLAB
that offers modeling, simulation, and analysis of dynamical
systems under a graphical user interface (GUI) environment.
• Simulink is based on block diagrams of Dynamic Systems
• The construction of a model is simplified with click-and-drag
mouse operations. Simulink includes a comprehensive block
library of toolboxes for both linear and nonlinear analyses.
• As Simulink is an integral part of MATLAB, it is easy to switch
back and forth during the analysis process and thus, the user
may take full advantage of features offered in both
environments.
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 2
Starting Simulink
• From Matlab command window enter simulink, or alternately,
click on the Simulink icon located on the toolbar
Simulink's library browser
Library Block set
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 3
Simulink library
• The block library is organized into
functional groups. For instance
• Sources: Blocks for the
generation of input signals
(steps, sinusoids etc.)
• Sinks: Blocks for the graphical
visualization of signals
• Math: Blocks for the
mathematical elaboration of
signals
• Continuous: Blocks for the
definition of (continuous)
transfer functions
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 4
Simulink – Sources Library
• I The most used blocks are:
• Constant: for generating a constant
value
• Step: for generating a step function
• Ramp: for generating a ramp
function
• Sine wave: for generating a
sinusoidal function
• From workspace: the reference
signal, previously generated in the
MATLAB workspace, is passed as
[time, value], where time
and value are two column
vectors with the same length
• Clock: it outputs the current
simulation time at each simulation
step
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 5
Simulink – Sinks Library
• Set of tools for displaying the output of the
simulations
• The most used blocks are:
• Scope: it displays the input signal as a
function of time (take care to the option
limit data points to…)
• XYGraph: it produce a graphics of the
signal y (on the second input) as a
function of the signal x (on the first input).
• To Workspace: it saves the samples of
the input signal in a MATLAB variable
(N.B.: save format array).
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 6
Simulink – Transfer functions
• In order to insert a transfer function in a Simulink scheme the blocks of
the library Continuous can be used:
• Transfer Fcn: it allows to define a transfer function by specifying
the vectors containing the coefficient of numerator and
denominator.
• Zero-Pole: it defines a transfer function by specifying the vectors
containing polse and zeros of the transfer function.
• The library Discrete contains the corresponding blocks for the
definition of discrete-time transfer functions.
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 7
Simulink and the Control System Toolbox
• The Control system Toolbox
provides a Simulink block for
directly using a transfer
function defined in the
MATLAB workspace (see
command tf) or a state space
model
Double click
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 8
Simulink – Simulation parameters
• Simulink solves ordinary differential equations (e.g. defined by
means of transfer functions) by using techniques for
numerical integration
• Simulating a dynamic system means computing the evolution
of the state and output variables for a given time period
• The state and the output values are computed at given time-
instants (called time steps) which are separeted by integration
intervals (called step sizes)
Computation of x(t), y(t)
Step size
X(0)
0=t0 t1 t2 t3 t4
Time step
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 9
Simulink – Methods for numerical integration
Several methods (the so-called solvers) for the numerical
integration of differential equations exist:
• Integration methods with fixed step size (useful for the
simulation of discrete-time systems)
• The smaller the step size is, the more accurate the
simulation results. On the other hand, the duration of the
simulation grows.
• Integration methods with variable step size
• the solver determines the optimal step size during the
simulation (for instance if the system is characterized by fast
dynamics, the step size must be reduced accordingly)
• ODE45 is the default solver
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 10
Simulation parameters
• Initial and final simulation time-instants
• Solver type (e.g. variable step, ODE45)
• Max step size (maximum size of the
interval beetwen two successive
time-steps)
Must be chosen:
• Smaller than the fastest time-constant
of the system
• Smaller than the period of the fastest
periodic signal acting on the system
(e.g. 1/10 smaller)
• Min step size (minimum size of the interval beetwen two successive
time-steps)
Can be used to reduce the simulation time (at the expense of the accuracy)
• Relative/absolute tollerance Accuracy of the simulation output (set
“auto” or values smaller enough, e.g. 1e-4)
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 11
Steps for constructing a Simulink model
• Open a new model from the Matlab menu file
or from the icon in the Simulink Library browser
Simulink workspace
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 12
Steps for constructing a Simulink model
• Import blocks in the Simulink workspace
Drag block to workspace window
• Connect blocks
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 13
Steps for constructing a Simulink model
• Action on blocks
Actions Keystrokes or Mouse Actions
Drag the block to the model window with the left
Copying a block from a library button on the mouse OR use select the COPY and
PASTE from EDIT menu.
Hold down the CTRL key, select the block with the
Duplicating blocks in a model left mouse button and drag the block to a new
location.
Display block's parameters Double click on the block
Flip a block CTRL‐I
Rotate a block (clockwise 90 deg @ each keystroke) CTRL‐R
Click on block's label and position the cursor to
Changing blocks' names
desired place.
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 14
Steps for constructing a Simulink model
• Save and run the simulation from the workspace window
or directly from MATLAB
>> sim(‘SimulinkModel’,EndTime)
• Display the results
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 15
Guidelines for a good simulation
• Matlab-Simulink models are usually structured in two main files:
M-file Simulink Model
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 16
Simulation of a feedback system - problem
With reference to the scheme of figure
1. Simulate the step response of the system with the continuous-
time transfer functions
In a unique figure plot the system's response
superimposed to the reference signal and the control
variable (two distinct subplots)
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 17
Simulation of a feedback system - problem
2. Simulate the step response of the feedback system with the
discrete-time transfer functions
and obtained from by discretization with sampling
time Ts = 0.1 s. Make the same plots of point 1.
3. Simulate the step response of the feedback system with the
discrete-time transfer functions R(z) and the continuous-time
plant G(s). Make the same plots of point 1.
4. Simulate the response of the system considered in the
previous point, by considering a reference signal computed
with the MATLAB function [q_t] = TrjPoly3(q0,q1,T,dt),
with q0=0, q1=1, T=2, dt = Ts.
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 18
Simulation of a differential equation
• A Simulink scheme allows the simulation of any differential
equation, even nonlinear or time-varying, if it is possible to
rewrite it an explicit form, i.e.
• The correpsonding block-scheme (to be designed in Simulink) is
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 19
Simulation of the simple pendulum
• The dynamics of the system is described by
the nonlinear differential equation
• In order to represent the equation with block-
schemes, it is convenient to rewrite it as
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 20
Simple pendulum – Simulink scheme
• From the equation
the Simulink scheme descends
w
To Workspace3
Cm
Constant
1 1
1/(M*l^2) q
s s
Manual Switch Integrator Integrator1 To Workspace
0
Gain
Constant1
Scope
Ci
To Workspace1 t
M*g*l sin Clock To Workspace2
Trigonometric
Gain1
Function
Parameters and initial conditions on and must be defined in
a separate m-file
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 21
Simple pendulum – Simulink scheme
• From the equation
the Simulink scheme descends
w
To Workspace3
Cm
Constant
1 1
1/(M*l^2) q
s s
Manual Switch Integrator Integrator1 To Workspace
0
Gain
Constant1
Scope
Ci
To Workspace1 t
M*g*l sin Clock To Workspace2
Trigonometric
Gain1
Function
In order to define the function it is possible to use other Simulink
blocks, in particular the blocks that allow the insertion of user-defined
functions
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 22
Blocks for inserting user-defined functions
Function directly
defined in the Simulink
block
External function
defined as m-function
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 23
Simple pendulum – alternatives schemes
w
To Workspace3
Cm
Constant
1 1
1/(M*l^2) q
s s
Manual Switch Integrator Integrator1 To Workspace
0
Gain
Constant1
Scope
Ci
Fcn1
To Workspace1 t
M*g*l*sin(u) Clock
To Workspace2
To Workspace3
Cm
Constant
1 1
1/(M*l^2) q
s s
Manual Switch Integrator Integrator1 To Workspace
0
Gain
Constant1
Scope
Ci
To Workspace1 MATLAB Fcn t
MATLAB
Clock
Function To Workspace2
function [Out] = PendFun(q) Global parameters in order to make them visible to
global M g l; the function (the same statement is present in the
Out = M*g*l*sin(q); m-file where the parameters are assigned)
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 24
Simple pendulum - problem
• Design a Simulink system for solving the equation of a simple
pendulum with friction, i.e.
• By assuming the parameters' values
simulate
• the free response from initial conditions
• the forced response to a constant input
• the complete response of the system
Plot in a unique figure (3 distinct subplots) the evolution of
in the three cases.
Luigi Biagiotti Systems and Control Theory Introduction to Simulink-- 25
Systems and Control Theory
Master Degree Course in ELECTRONICS ENGINEERING
http://www.dii.unimore.it/~lbiagiotti/SystemsControlTheory.html
Introduction to SIMULINK
Luigi Biagiotti
e-mail: luigi.biagiotti@unimore.it
http://www.dii.unimore.it/~lbiagiotti
