Introduction to Control Theory Including Optimal Control
Nguyen Tan Tien - 2002.3
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1. System Dynamics and Differential Equations
1.1 Introduction
1.4 shows a tank with: inflow rate q i , outflow rate q o , head
- Investigation of dynamics behavior of idealized components,
and the collection of interacting components or systems are
equivalent to studying differential equations.
- Dynamics behavior of systems ⇔ differential equations.
level h , and the cross-section are of the tank is A .
inflow
1.2 Some System Equations
h (head)
Inductance circuit: In electromagnetic theory it is known that,
for a coil, in Fig.1.1, having an inductance L , the
electromotive force (e.m.f.) E is proportional to the rate of
change of the current I , at the instant considered, that is,
E=L
dI
dt
or I =
∫
1
E dt
L
(1.1)
L
valve
Fig. 1.4
If q = q o − q i is the net rate of inflow into a tank, over a
period δ t , and δ h is the corresponding change in the head
level, then q δ t = A δ h and in the limit as δ t → 0
q=A
E
Fig. 1.1
Capacitance circuit: Similarly, from electrostatics theory we
know, in Fig. 1.2, that the voltage V and current I through a
capacitor of capacitance C are related at the instant t by
∫ I dt
dE
or I = C
dt
or h =
1
A
∫ q dt
(1.4)
dy
=x
dt
(1.5)
(1.2)
Equation (1.5) is interesting because:
- its solution is also the solution to any one of the systems
considered.
- it shows the direct analogies which can be formulate
between quite different types of components and systems.
- it has very important implications in mathematical
modeling because solving a differential equation leads to
the solution of a vast number of problems in different
disciplines, all of which are modeled by the same equation.
C
E
Fig. 1.2
Dashpot device: Fig. 1.3 illustrates a dashpot device which
consists of a piston sliding in an oil filled cylinder. The motion
of the piston relate to the cylinder is resisted by the oil, and
this viscous drag can be assumed to be proportional to the
velocity of the piston.
If the applied force is f (t ) and the corresponding displacement
is y (t ) , then the Newton’s Law of motion is
f =µ
dh
dt
All these equations (1.1)-(1.4) have something in common:
they can all be written in the form
a
1
E=
C
discharge q0
dy
dt
(1.3)
where the mass of the piston is considered negligible, and µ is
the viscous damping coefficient.
y (t )
Over small ranges of the variables involved the loss of
accuracy may be very small and the simplification of
calculations may be great. It is known that the flow rate
through a restriction such as a discharge valve is of the form
q =V
P
where P is the pressure across the valve and V is coefficient
dependent on the properties of the liquid and the geometry of
the valve. Fig.1.5 shows the relation between the pressure and
the flow rate.
q flow rate
q1
assumed
linear
f (t )
Fig.1.3
Liquid level: To analyze the control of the level of a liquid in
a tank we must consider the input and output regulations. Fig.
P1
P (pressure)
Fig. 1.5
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1
Chapter 1 System Dynamics and Differential Equation
Introduction to Control Theory Including Optimal Control
Nguyen Tan Tien - 2002.3
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Assume that in the neighborhood of the pressure P = P1
We can also write the set of simultaneous first order equation
(1.7) and (1.8) as one differential equation of the second order.
On differentiating (1.7), we obtain
change in pressure
≈ R'
change in flow rate
h&&2 =
where R' is a constant called the resistance of the valve at the
point considered.
This type of assumption, which assumes that an inherently
nonlinear situation can be approximated by a linear one albeit
in a restricted range of the variable, is fundamental to many
applications of control theory.
1
A
1
1 ⎛ 1
1 ⎞
1
⎜
⎟ h2 +
+
q − 2 h&2 −
h2
A1
A1
A1 ⎜⎝ R1 R 2 ⎟⎠
A1 R1
A
1
1
=
q − 2 h&2 −
h2
A1
A1
A1 R 2
Then
h&&2 =
=
2
A2
A1
h1
R1
h2
⎡ 1
1
q−⎢
A1 A2 R1
⎢⎣ A2
⎡ 1
h&&2 + ⎢
⎣⎢ A2
⎛ 1
1
⎜
⎜R + R
2
⎝ 1
dh1
1
= q−
(h1 − h2 )
dt
R1
⎞
1 ⎤&
1
⎟+
⎟ A R ⎥ h2 − A A R R h2
1 1 ⎥⎦
1 2 1 2
⎠
⎛ 1
1
⎜
⎜R +R
2
⎝ 1
Consider a control system: a tank with an inflow q i and a
(1.7)
discharge q o in Fig. 1.7
For tank 2
⎛ 1
1
⎜
⎜R + R
2
⎝ 1
float
lever
dh
1
1
(h1 − h2 ) −
h2
A2 2 =
dt
R1
R2
⇒
⎞&
⎟ h2
⎟
⎠
1.3 System Control
dh
1
1
1
h&1 = 1 =
q−
h1 +
h2
dt
A1
A1 R1
A1 R1
dh
1
1
h&2 = 2 =
h1 −
dt
A2 R1
A2
⎛ 1
1
⎜
⎜R + R
2
⎝ 1
⎞
1 ⎤&
1
1
⎟+
⎟ A R ⎥ h 2 + A A R R h2 = A A R q
1 1 ⎦⎥
1 2 1 2
1 2 1
⎠
(1.11)
Equation (1.11) is a differential equation of order 2 of the
form &y& + a1 y& + a 2 y = b0 x .
R2
For tank 1
⇒
1
1 &
1
1
q−
h2 −
h2 −
A1 A2 R1
A1 R1
A1 A2 R1 R2
A2
or
Fig. 1.6
A1
(1.10)
=
inflow
q
⎞&
⎟ h2
⎟
⎠
1
1
1
h&1 =
q−
h1 +
h2
A1
A1 R1
A1 R1
(1.6)
A two tank systems with one inflow (q) and two discharge
valves is shown in Fig. 1.6.
⎛ 1
1
⎜
⎜R + R
2
⎝ 1
From (1.7) and (1.8),
At a given point, the pressure P = hρ g , define R = R' /( ρ g ) ,
we can rewrite the above relation as
R = h / qo
1 &
1
h1 −
A2 R1
A2
⎞
⎟ h2
⎟
⎠
R
(1.8)
inflow
outflow
regulator valve
Fig. 1.7
We can write (1.7) and (1.8) in matrix form as follows
1
⎡
−
⎡ h&1 ⎤ ⎢⎢ A1 R1
⎢& ⎥ = ⎢ 1
1
⎣ h2 ⎦
−
⎢ A R
A2
⎣ 2 1
1
⎤
⎡ 1 ⎤
⎥
A1 R1
⎥ ⎡ h1 ⎤ + ⎢ A ⎥ q
⎢
⎥
⎞
⎛ 1
h
1 ⎥⎣ 2⎦ ⎢ 1⎥
⎟
⎜
⎣ 0 ⎦
⎜ R + R ⎟⎥
2 ⎠⎦
⎝ 1
that is, in the form
h& = A h + B q
(1.9)
where
1
⎡
⎢− A R
⎡ h&1 ⎤
1
1
h= ⎢& ⎥ , A= ⎢
1
1
⎢
h
⎣ 2⎦
−
⎢ A R
A
2
1
2
⎣
1
⎤
⎡1 ⎤
⎥
A1 R1
⎥, B=⎢A ⎥
⎛ 1
⎢ 1⎥
1 ⎞⎥
⎟
⎜
⎣ 0 ⎦
⎜ R + R ⎟⎥
2 ⎠⎦
⎝ 1
The purpose of controller is to maintain the desired level h .
The control system works very simply. Suppose for some
reason, the level of the liquid in the tank rises above h . The
float senses this rise, and communicates it via the lever to the
regulating valve which reduces or stops the inflow rate. This
situation will continue until the level of the liquid again
stabilizes at its steady state h .
The above illustration is a simple example of a feedback
control system. We can illustrate the situation schematically
by use of a block diagram as in Fig. 1.8.
ε
h1
x
regulator
plant
h2
h2
Fig. 1.8
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2
Chapter 1 System Dynamics and Differential Equation
Introduction to Control Theory Including Optimal Control
Nguyen Tan Tien - 2002.3
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
theory is to design – in terms of the transfer function – a
system which satisfies certain assigned specifications. This
objective is primarily achieved by a trial and error approach.
with
h1
: desired value of the head
h2
: actual value of head
ε = h1 − h2 : error
The regulator receives the signal ε which transforms into a
movement x of the lever which in turn influences the position
of the regulator valve.
To obtain the dynamics characteristics of this system we
consider a deviation ε from the desired level h1 over a short
time interval dt .
If q i and q o are the change in the inflow and outflow, from
(1.4),
A
dε
= qi − q o
dt
where (1.6), q o = ε / R . So that the equation can be written as
AR
Modern control theory is not only applicable to linear
autonomous systems but also to time-varying systems and it is
useful when dealing with nonlinear systems. In particularly it
is applicable to MIMO systems – in contrast to classical
control theory. The approach is based on the concept of state.
It is the consequence of an important characteristic of a
dynamical system, namely that its instantaneous behavior is
dependent on its past history, so that the behavior of the
system at time t > t 0 can be determined given
(1) the forcing function (that is, the input), and
(2) the state of the system at t = t 0
In contrast to the trial and error approach of classical
control, it is often possible in modern control theory to
make use of optimal methods.
dε
+ ε = R qi
dt
the change in the inflow, depends on the characteristics of the
regulator valve and may be of the form
q i = K ε , K is a constant
Another type of control system is known as an open loop
control system, in which the output of the system is not
involved in its control. Basically the control on the plant is
exercised by a controller as shown in Fig. 1.9.
input
output
controller
plant
Fig. 1.9
1.4 Mathematical Models and Differential Equations
Many dynamic systems are characterized by differential
equations. The process involved, that is, the use of physical
laws together with various assumptions of linearity, etc., is
known as mathematical modeling.
Linear differential equation
&y& + 2 y& − 3 y = u → time-invariant (autonomous) system
&y& − 2 t y& + y = u → time-varying (non-autonomous) system
Nonlinear differential equation
&y& + 2 y& 2 − 2 y = u
&y& − y 2 y& + y = u
1.5 The Classical and Modern Control Theory
Classical control theory is based on Laplace transforms and
applies to linear autonomous systems with SISO. A function
called transfer function relating the input-output relationship
of the system is defined. One of the objectives of control
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Chapter 1 System Dynamics and Differential Equation