Chapter 1
Differential equations
represented as block
diagrams
1.1
Introduction
The basic mathematical model form of a dynamic system is the differential
equation since applying physical principles results in one of more
differential equation describing the system. A given differential equation
can be represented conveniently in two alternative forms:
• Block diagrams: A (mathematical) block diagram gives a graphical
representation of a mathematical model, and therefore gives good
information of the structure of the model, e.g. how subsystems are
connected. With block diagrams you can build simulators in
graphical simulation tools as SIMULINK and LabVIEW Simulation
Module.
• State-space models: State-space models are just a structured way
to write the differential equations for a system: All the
time-derivatives are of first order, and they appear on the left-hand
side of the differential equations. Various tools for analysis and
design and simulaton of dynamic systems and control systems
assume state-space models.
In the following the block diagram representation is explained. State-space
models are not relevant for the Project part of the course Control with
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Implementation.
1.2
1.2.1
Mathematical block diagrams
Commonly used blocks in block diagrams
Figure 1.1 shows the most frequently used blocks, which we can call the
elementary blocks, that we can use for drawing block diagrams.
Name:
Symbol:
Function:
y0
Integrator
u=y
y
t
y(t) = y 0 +
u(t)dt
0
u1
Sum (incl. subtraction )
(No sign means plus .)
u2
y
y = u1 + u2 – u3
u3
u1
Multiplication
MULT
y
y = u1u2
u2
u1
Division
Gain
Time delay
y
DIV
u2
u
u
K
y
y
y = u1/u2
y = Ku
y(t) = Ku
Figure 1.1: Elementary blocks for drawing block diagrams
Other blocks than the elementary blocks shown in Figure 1.1 must be used
to represent non-linear functions. Figure 1.2 shows a few such blocks. You
can define the function and the appearance of a block yourself.
Finn Haugen: SCE1106 Control with Implementation
Saturation
Rate limiter
Dead zone
Relay
u
y
u
y
u
y
u
y
u1
Switch
Control
signal, c
3
y
u2
Figure 1.2: Blocks for non-linear functions
1.2.2
How to draw a block diagram
A systematic way of drawing the block diagram of a given differential
equation is as follows:
1. Write the differential equations so that the variable of the highest
time-derivative order appears alone on the left part of the equations.
2. For first order differential equations (first order time-derivatives):
Draw one integrator for the each of the variables that appear with its
time-derivative in the model, so that the time-derivative is at the
integrator input and the variable itself is at the integrator output.
These variables, at the integrator outputs, are denoted
state-variables, because their values at any instant of time represent
the state of the system. The general names of the state-variables are
x1 , x2 etc. The initial values of the state-variables are the initial
outputs of the integrator.
For second order differential equations (second order
time-derivatives): Draw two integrators (from left to right) and
connect them in series. The second order time-derivative is then the
input to the leftmost integrator. (The variables at the integrator
outputs are state-variables.)
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You will probably not see third or higher order differential equations,
but if you do, the above procedure is naturally extended.
3. Connect the integrators together according to the differential
equation using proper blocks, cf. Figures 1.1 and 1.2.
The following example demonstrates the above procedure.
Example 1.1 Block diagram of mass-spring-damper system
Figure 1.3 shows a mass-spring-damper-system. y is position. F is applied
K [N/m]
F [N]
m
D [N/(m/s)]
0
y [m]
Figure 1.3: Mass-spring-damper
force. D is damping constant. K is spring constant. It is assumed that the
damping force Fd is proportional to the velocity, and that the spring force
Fs is proportional to the position of the mass. The spring force is assumed
to be zero when y is zero. Force balance (Newtons 2. Law) yields1
mÿ(t) = F (t) − Fd (t) − Fs (t)
= F (t) − Dẏ(t) − Ky(t)
(1.1)
which is a second order differential equation.
We isolate ÿ at the left side:
ÿ(t) =
1
[F (t) − Dẏ(t) − Ky(t)]
m
(1.2)
Then we draw two integrators and connect them in series. ÿ is the input to
the first (leftmost) integrator. Finally, we complete the block diagram
according to the model. The resulting block diagram is shown in Figure
1.4. The state-variables x1 and x2 are defined as y and ẏ, respectively.
[End of Example 1.1]
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Double-dot is used for second order time-derivative: ÿ(t) ≡ d2 y(t)/dt2
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my = F - Dy - Ky
F
Sum
m
x
DIV
÷
y(0)
y(0)
y
y
y
x1
x2
Integrator
Integrator
Dy
MULT
D
Ky
MULT
K
Figure 1.4: Block diagram of (1.2)
1.2.3
Simulators based on block diagram models
LabVIEW, SIMULINK, and Scicos are examples of simulation tools for
block diagram models.2 Figure 1.5 shows the block diagram of the
mass-spring-damper system in LabVIEW, and Figure 1.6 shows the front
panel (the user interface) of the simulator. The front panel contains
adjustable elements (controls), and indicators displaying resulting values
and plots.
Figure 1.7 shows a SIMULINK block diagram of the mass-spring-damper
system. The simulated force u and the response in the position y due to a
step in the force will be shown in respective scopes. Since the responses are
the same as shown in Figure 1.6, the scopes are not depicted here.
2
Tutorials are available at http://techteach.no.
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Figure 1.5: The block diagram of the mass-spring-damper system simulator in
LabVIEW
Figure 1.6: The front panel of the mass-spring-damper system simulator in
LabVIEW
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Figure 1.7: SIMULINK block diagram of the mass-spring-damper system
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