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1
Lecture Course in
Adaptive Control
for the 4th
Class of Control Engineering in the
Control and Systems Engineering
Department at the University of
TechnologyCCE-CN445
Prepared By:
Dr. Mohammed Y. Hassan, Ph. D.
Adaptive Control
2
Syllabus
Fourth Year-Control Eng. Theoretical: 2 Hr./ Week
CCE-CN445 (Semester)
1. Introduction and General Aspects: (4 Hrs)
What is Adaptive Control?
Relationship between Non-Adaptive, robust and Adaptive Control.
Performance objectives and design constraints for the control
engineers.
2. Types of Adaptive Control strategies: (2 Hrs)
Gain Scheduling Regulators
Self-Tuning Regulators.
Model Reference Adaptive Control.
3. Gain Scheduling Regulator: (2 Hrs)
4. Self Tuning Regulator: (8 Hrs)
Pole-Assignment by output Feedback.
5. Model Reference Adaptive Control: (6 Hrs)
6. Practical aspects and Implementation: (2 Hrs)
7. Relation to other topics: (6 Hrs)
Expert Systems.
Neuro Controller.
Fuzzy Controller.
Genetic Auto-Tuners.
Total: 30 Hrs
Adaptive Control
3
References:
1. “Adaptive Control”
By: K. J. ÅstrÖm and B. Wittenmark.
2. “Self Tuning Systems”
By: P. E. Wellstead and M. B. Zarrop.
Adaptive Control
4
ADAPTIVE CONTROL
1-Introduction
1.1-Definitions:
Adaptive controller:
Is a controller that can modify its behavior in response to changes in the
dynamics of the process and the disturbances.
Adaptive system:
Is any physical system that has been designed with an adaptive view
point.
1.2 Historical background:
In the early 1950, there was extensive research on adaptive control, in
connection with the design of autopilots for high performance aircraft. Such
aircraft operate over a wide range of speeds and altitudes. It was found that
ordinary constant-gain, linear feedback control could work well in one
operating condition, but that changed operating conditions led to difficulties.
A more sophisticated regulator, which could work well over a wide range of
operating conditions was therefore needed.
In the 1960s many contributions to control theory were important for the
development of adaptive control. State space and stability theory were
introduced. There were also important results in stochastic control theory.
Dynamic programming introduced by Bellman, increased the understanding of
Adaptive Control
5
adaptive processes. Fundamental contributions were also made by Tsypkin,
who showed that many schemes for learning and adaptive control could be
described in a common framework as recursive equations of a special type.
There were also major developments in system identification and in parameter
estimation.
In the late 1970s and early 1980s correct proofs for stability of adaptive
systems appeared. Rapid and revolutionary progress in microelectronics has
made it possible to implement adaptive regulators simply and cheaply.
Vigorous development of the field is now taking place, both on universities and
industry. Several commercial adaptive regulators based on different ideas are
appearing on the market. A great number of industrial control loops are under
adaptive control. These include a wide range of applications in aerospace,
process control, ship steering, robotics and other industrial control systems.
To works with adaptive control one must have a background in
conventional feedback control and also sampled data systems. The reason for
this is that virtually all adaptive systems are implemented using digital
computers.
However, adaptive control has links in many directions, some of which are
illustrated in figure bellow:
Adaptive Control
6
Fig. Links with adaptive control.
There are strong ties to nonlinear system theory, because adaptive systems are
inherently nonlinear.
1.3 Adaptation and tuning:
It is customary to separate the tuning and adaptation problems. In the
tuning problem it is assumed that the process to be controlled has constant but
unknown parameters; in the adaptation problem it is assumed that the
parameters are changing.
Adaptive Control
7
Many issues are much easier to handle in the tuning problem. The convergence
problem is to investigate where the parameters converge to their true values.
The corresponding problem is much more difficult in the adaptive case,
because true values are changing.
1.4 Direct and Indirect adaptive controllers:
In general, the adaptive controllers can be divided into two algorithms;
direct and indirect. In direct algorithms, the parameters are updated directly.
If the controller parameters are obtained indirectly via a design procedure, we
use the term indirect algorithms. Sometimes, it is possible to re-
parameterize the process model such that it is possible to use either a direct or
indirect controller.
However, the indirect methods have sometimes been called explicit self-
tuning control, since the process parameters have been estimated. Direct
updating of the regulator parameters has been called implicit self tuning
control.
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8
2-Why adaptive control:
There are many reasons for adaptive control. The key factors are:
a-Variations in process dynamics. Parameters may vary due to nonlinear
actuators, changes in the operating conditions of the process, and non-
satisfactory disturbances acting on the process.
b- Variations in the character of the disturbances.
c- Engineering efficiency.
However, the most common regulators is a feedback controller with
fixed parameters. Through feedback it is possible to decrease the sensitivity to
parameter variations by increasing the loop gain of the system. The main
drawbacks of high-gain controllers are the magnitude of the control signal and
the problem of stability of the closed loop system. If there are bounds on the
uncertainties of the process parameters, it is possible to design robust
controllers by increasing the complexity of the controller. To use this
approach, it is necessary to know the structure of the process fairly accurately
and to have bounds on the variations of the parameters.
In the following, two examples in which parameter variations cause
difficulties when a fixed gain regulator is used:
Example 1: Nonlinear actuators (nonlinear valve):
Many actuators have a nonlinear characteristics which creates
difficulties unless special precautions are taken.
A simple feedback loop with a nonlinear valve is shown in figure below:
Adaptive Control
9
The static valve characteristics be:
u)u(fV4
0u
Linearizing the system around a steady state operating point shows that the
loop gain is proportional to )u(f
. It then follows that the system can perform
well at one operating point and poorly at another. This is illustrated by the step
responses for PI control with k=0.15, Ti=1 and
)1s(
1)s(G
3
.
Fig. Step responses for PI control of the simple flow loop.
Adaptive Control
10
Example 2: Airplane control (Short-period airplane dynamics):
Control of high performance airplane was strong driving force for the
development of adaptive control in 1960s. The dynamics of an airplane depend
on the speed, altitude, angle of attack, …etc. Also, the flexible structure of the
airplane will cause difficulties when designing the control system. In figure
below, the notations for the airplane is given:
The rigid body or the so-called short period dynamics for an experimental F4-
E with canards will be given. The canards make it easier to maneuver the
airplane at the cost of stability. The dynamics can be linearized around
stationary flight conditions (i.e. constant speed and altitude and small angle of
attack α). Let the state variables be normal acceleration (Nz) and pitch rate q=
.
. It is assumed that the dynamics of the servo of the elevons and canards have
the transfer function:
as
a)s(G
with a=14
Let δe be the corresponding state. The model of the airplane has three states
Nz, q, and δe and the linearized equations have the form:
u
a
0
1b
X
a00
23a22a21a
13a12a11a
X.
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11
Assuming the eigen values of the short-period dynamics are λ1 and λ2, it is
seen that the airplane is unstable for subsonic speed and poorly damped for
supersonic speeds as shown in the table below:
mach 0.5 0.9 1.5
Altitude (feet) 5000 35000 35000
a11 -0.9896 -0.667 -0.5162
a12 17.41 18.11 26.96
a13 96.15 84.34 178.9
a21 0.2648 0.08201 -0.6896
a22 -0.8512 -0.6587 -1.225
a23 -11.39 -10.81 -30.38
b1 -97.78 -85.09 -175.6
λ1 -3.07 -1.87 -0.87-43i
λ2 1.23 0.56 -0.87+4.3i
Note: It is essential to know the frequency response at the desired crossover
frequency in order judge whether parameter variation (position of poles) will
have any effect on the closed loop system properties. Step responses are poor
guide; it is much better to look at the frequency responses.
3- Adaptive schemes:
There are three main adaptive schemes:
a-Gain scheduling.
b-Self-tuning regulator (STR).
c-Model-reference adaptive control (MRAC).
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12
a-Gain scheduling:
In many situations it is known how the dynamics of a process change
with the operating conditions of the process. One source for the change in
dynamics may be nonlinearities that are known. It is then possible to change
the parameters of the controller by monitoring the operating conditions of the
process. This idea is called gain scheduling.
Its principle is to reduce the effects of parameter variations by
changing the parameters of the regulator as function of auxiliary variables that
correlate well with these changes in process dynamics. It is a nonlinear
feedback controller of a special type. It has a linear regulator whose
parameters are changed as a function of operating conditions in a
programmed way.
Gain scheduling was used in special cases: such as autopilots for high-
performance air-craft.
The principle:
It is sometimes possible to find auxiliary variable that correlate
well with the changes in process dynamics. It is then to reduce the effects of
parameter variations simply by changing the parameters of the regulator as
functions of auxiliary variables as shown in figure below:
Adaptive Control
13
Gain scheduling can thus be viewed as a feedback control system in which the
feedback gains are adjusted using feed forward compensation.
A main problem in the design of systems with gain scheduling is to find
suitable scheduling variables. This is normally done based on knowledge on the
physics of a system. When scheduling variables have been determined, the
regulator parameters are calculated at a number of operating conditions, using
some suitable design method. The regulator is thus tuned or calibrated for
each operating condition. The stability and performance of the system are
typically evaluated by simulation.
Gain scheduling has the advantage that the regulator parameters can be
changed quickly in response to process changes.
However, gain scheduling has two main disadvantages:
1) It is time-consuming to find the correct scheduling which should be
checked by extensive simulation.
2) There is no feedback to compensate for incorrect scheduling.
Design of Gain Scheduling regulators:
It is difficult to give general rules for designing gain scheduling
regulators. The key question is to determine the variables that can be used as
scheduling variables. It is clear that these auxiliary signals must reflect the
operating conditions of the plant. The following general ideas can be useful:
1) Linearization of nonlinear actuators.
2) Gain scheduling based on measurements of auxiliary variables.
3) Time scaling based on production rate.
4) Nonlinear transformation.
Adaptive Control
14
Example 1: Nonlinear Actuator:
Consider the system with a nonlinear valve characteristics
discussed previously. The nonlinearity is assumed to be:
u)u(fV4
0u
Let f1 be an approximation of the inverse of the valve characteristics. To
compensate for the nonlinearity; the output of the regulator is fed through this
function before it is applied to the valve. This gives the relation:
))c(f(f)u(fV 1
where c is the output of the PI regulator. The function ))c(f(f 1 should have
less variation in gain the f. If f1 is the exact inverse, then v=c. Assume that
f(u) is approximated by two straight lines: one connecting the point (0,0) and
(1.3,3), and the other connecting (1.3,3) and (2,16) as shown in figure below:
Then:
16C3..............
3C0.............
139.1C0538.0
C433.0)c(f
1
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15
The figure below shows step changes in the reference signal at three different
operating conditions when the approximation of the inverse of the valve
characteristics is used between the regulator and valve.
By improving the inverse it is possible to make the process even more
insensitive to nonlinearity of the valve.
Example 2: Tank System:
Consider a tank where the cross section A varies with height h.
The model is:
h.g.2.aqi)h),h(A(dt
d
Where qi is the input flow and a is the cross section of the outlet pipe. Let qi be
the input and h is the output of the system. The schematic diagram of the tank
is shown in figure below:
Adaptive Control
16
The linearized model at an operating point qin and h, is given by the transfer
function:
s)s(G
where : )h(A
1o
and oo
o
oo
o
h).h(A.2
h.g.2.a
h).h(A.2
qin
A good PI control of the tank is given by:
)d)(eTi
1)t(e(K)t(u
where
2K and
2
2Ti
Introducing the expressions for α and β gives the following gain schedule:
o
o
o
h.2
qin)h(A...2K
2oo
o
i
.h).h(A.2
qi.2T
The numerical values are often such that α << 2ζω. The schedule can then be
simplified to: K=2ζωA(h) and Ti=
2
In this case it is sufficient to make the gain proportional to the cross section of
the tank. For ζ=0.5 and ω=1 rad/sec, the gain schedule of the system is:
Adaptive Control
17
Cross section A(h), (m2) Gain (K)
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
From the two examples, it was possible to determine the schedules exactly. The
behavior of the closed loop system does not depend on the operating
conditions. In other cases it is possible to obtain only approximate relations for
different operating conditions. The design then has to be repeated for several
operating conditions to create a table. The gain schedule is usually obtained
through simulations of a process model, but it is possible to build up the gain
table online. This might be done by using an auto-tuner or an adaptive
controller. The adaptive system is used to get the controller parameters for
different operating points. The parameters are then stored for later use when
the system returns to the same or a neighboring operating point.
H.w.: Read the other examples discussed in chapter nine.
Adaptive Control
18
b- Self-Tuning regulator:
Basic idea:
In an adaptive system, it is assumed that the regulator parameters are adjusted
all the time. This implies that the regulator parameters follow changes in
process. It is difficult to analyze the convergence and stability properties of
such systems. To simplify the problem it can assume that the process has
constant but unknown parameters. When the process is known, the design
procedure specifies a set of desired controller parameters. The adaptive
controller should converge to these parameter values even when the process is
known. A regulator with this property is called Self-Tuning, since it
automatically tunes the controller to the desired performance.
The Self- Tuning Regulator (STR) is based on the idea of separating the
estimation of unknown parameters from the design of the controller. The basic
idea is illustrated in figure bellow:
Fig. Block Diagram of Self Tuning Regulator (STR)
The Unknown parameters are estimated on-line using Recursive estimation
method. The estimated parameters are tested as if they are true. i. e., the
uncertainties of the estimates are not considered.
Adaptive Control
19
Many different estimation schemes can be used, such as:
a) Stochastic approximation.
b) Least squares.
c) Extended and generalized least squares.
d) Instrumental variable.
e) Maximum likelihood.
In the block diagram, the design block represents an on-line solution to the
design problem for a system with unknown parameters. This is called the
underlying design problem. Examples of the design methods can be used are:
a) Minimum variance.
b) Linear quadratic (LQ).
c) Pole placement.
d) Model following.
The design method is chosen depending on the specifications of the closed loop
systems. Different combinations of estimation methods lead to regulators with
different properties.
However, Self Tuning regulators were originally derived for sampled data
systems but continues time and hybrid algorithms have also benn developed.
Moreover, Self Tuning regulators are based on the idea of estimating some
parameters of the process. The most straight forward approach is to estimate
the parameters of the transfer function of the process and the disturbances.
This gives an indirect adaptive algorithm. The regulator parameters are not
updated directly, but rather indirectly via the estimation of the process model.
This type of adaptive control, base com least squares estimation and deadbeat
control.
It is possible to re-parameterize the model in the regulator parameters such
that the regulator parameters can be estimated directly. That is a direct
adaptive control.
Adaptive Control
20
Disadvantages:
There are three main disadvantages:
1- The stability analysis is complicated because the regulator parameters
depend on the estimated parameters in complicated way.
2- The Map from the process parameters to regulator parameters may have
singular points. This happens in design methods based on pole placement.
3- To ensure that the parameters converge to the correct values, it is
necessary that the model structure be correct and that the process input be
persistently exciting.
In order to form a Self tuning or adaptive control system, three forms of
controls can be used:
1- Pole placement (assignment) control.
2- Minimum variance control.
3- Multistage predictive control.
In this lecture, the pole placement control will be taken into consideration.
Pole assignment control:
It is a classical approach to controller specifications. The design of a
feedback controller has two main aims:
1- the first is to modify in some way the dynamic response of a system.
2- The second os to reduce the sensitivity of the system output to
disturbances.
Suppose we have a system with a continuous time transfer function:
)s(us.1
f)s(Y
(1)
where is the time constant of the system and f is the system gain.
The discrete-time domain transfer function is:
Adaptive Control
21
)z(UZ.a1
b.Z)z(Y
1
1
(2)
where
T
ea and f).a1(b (3)
and T is the sampling time.
Common control objective would be to use a feedback controller to alter the
speed of the system response. I.e. the original system might have a certain
speed of response to a sudden change in u(z) associated with time constant ,
but it is required to change this response rate to one associated with time
constant .
In terms of the system model, equation 2 changing the speed of response
corresponds to later the value of a in the denominator in some way assigning it
another value corresponding to faster speed of response.
Also, it is required to ensure that the output y(t) come into correspondence
with reference signal r(t).
To do this, propose the feedback controller structure with the following block
diagram is:
Adaptive Control
22
Where:
u(Z)=-g.y(Z)+h.r(Z) (4)
By combining Eq. (2) and Eq. (4), this gives the closed loop transfer function:
)Z(rZ).g.ba(1
Z.h.b)z(y
1
1
(5)
Notice that the closed loop speed of structure is determined by (a-b.g) instead
of a.
In other words, the open loop system pole Z=a has been assigned (placed) to a
closed loop position Z=a-b.g. Hence, we can specify the location of the poles as
a design parameter.
Suppose we required that closed loop pole at Z=t1 , then:
g.bat1
(6)
or
b
tag 1
(Design equation) (7)
The remaining controller objective is to ensure that steady state (when r(z) is
constant), the output y(z)=r(z).
This is done by selecting the controller parameter h such that the closed loop
transfer function is equal to one at zero frequency. (Zero frequency means
0s or 1Z ). Using equation 5:
Adaptive Control
23
1)g.ba(1
h.b
(8)
or b
)g.ba(1h
. (9)
Equation (9) is the design equation for selecting h so that y(z) tracks r(z) at
steady state conditions.
The key idea of pole assignment is to shift the open loop poles to some desired
set of closed loop poles.
Adaptive Control
24
Example:
Design a digital controller for the open loop discrete transfer function of
1Z
2)z(GOL
, so that the digital closed loop transfer function has a zero steady
state error with characteristic equation of 1Z.5.01T
.
Solution:
The open loop transfer function can be re-written as: 1
1
OLZ1
Z2)z(G
The closed loop transfer function is shown below:
)z(rZ).g.21(1
Z.h.2)z(y
1
1
By equating c\cs. equation with T, this gives:
1-2.g=0.5
2g=0.5
g=0.25
For zero steady state error,
5.01
h.21
)z(r
)z(ylim
1Z
2.h=1-0.5
2.h=0.5
h=0.25
Adaptive Control
25
The new block diagram of closed loop transfer function is:
and
H.W.:
Obtain T in s-plane if the sampling time is 1 second.
)z(rZ.5.01
Z.5.0)z(y
1
1
Adaptive Control
26
General Algorithm of pole assignment:
Consider a system defined by the equation:
A.y(z)=B.u(z)+C.e(z) (10)
Where y(z) is the output, u(z) is the input and e(z) is the disturbance.
Consider a noise-free system model:
A.y(z)=B.u(z) (11)
With a controller of the form:
F.u(z)=-G.y(z)+H.r(z)
Where:
nf
nf
1
.1 Z.f..............Zf1F
ng
ng
1
.1O Z.g..............ZggG
nh
nh
1
.1O Z.h..............ZhhH
Then, the closed loop system equation is, (combining equation 10 and 11):
(F.A+B.G)y(z)=B.H.r(z)
(12)
Fig. Closed loop system with controller.
Now, if the desired closed loop pole set is defined by the roots of:
nt
nt
2
2
1
1 Z.t.......Z.tZ.t1T
)z(rG.BA.F
H.B)z(y
Adaptive Control
27
then the controller coefficients which assign the actual pole set to the desired
set are given by the solution of:
(13)
Equation (13) can be written as a set of simultaneous equations by equating
coefficients of like power of iZ
.If the order nf and ng are chosen correctly,
then the set of simultaneous equations will have a unique solution.
Re-writing equation (13) in terms of the polynomial gives:
)Z.f..............Zf1(nf
nf
1
.1
. )Z.a..............Za1(na
na
1
.1
.+
)Z.b..............Zb(nb
nb
1
.1
. )Z.g..............Zgg(ng
ng
1
.1O
=
nt
nt
2
2
1
1 Z.t.......Z.tZ.t1 (14)
The above equation shows that:
a) Because the largest power of iZ
to occur on the L.H.S are (nf + na) and
(ng + nb), then it is possible to generate, (by comparing coefficients of iZ
), nf + na= ng + nb equations.
b) There are 1+ nf + ng unknown coefficients of the polynomials F and G.
Hence, for a unique solution to equation (13), we require:
nf + na= ng + nb= nf + ng +1
or
Conditions of the design
F.A+B.G=T
nf = nb-1
ng = na -1
nt na+ nb-1
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28
Solution of equation (13)
Equation (13) is called the pole placement identity. It can be solved in a
number of ways. A direct solution has obtained by comparing coefficients of
iZ
leads to a matrix \vector equation with a special banded structure.
Equation (14) becomes:
)1na(
.1.na
3
.1.2
2
.1.1
1
.1
na
na
2
.2
1
.1 Z.fa..............ZfaZfaZfZ.a..............ZaZa1
)nfna(
nfna
)1nf(
nf1
nf
.nf
)2na(
.2.na
3
.2.1
2
.2 Z.f.a......Z.f.aZf...........Zfa....................ZfaZf
.......Z.gb.............ZgbZgbZ.gb.............ZgbZgb)2ng(
.ng.2
3
.1.2
2
o.2
)1ng(
.ng.1
2
.1.1
1
o.1
.
.
.
nt
nt
2
2
1
1
)nbng(
.ngnb
)1nb(
.1nb
nb
o.nb Z.t........Z.tZ.t1Z.gb.............ZgbZgb
By equating coefficients of )nfna(21Z..............Z,Z
or )nbng(Z
, we get:
1o111 tg.bfa power of 1Z
211o22112 tg.bg.bff.aa power of 2Z
.
.
.
and so on.
In matrix form, we get:
ng
1
o
nf
2
1
nbna
2nb
1nb1na
1
121
1
g
.
g
g
f
.
.
.
f
f
.
b..000a...0
...........
...........
b...b0....0
b....ba...a
......1....
...........
.........a.
0...bb0.01a
0...0b0..01
=
0
.
0
0
.
.
.
.
at
at
22
11
(15)
A c b
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29
A.c=b
Where:
c is a vector with controller coefficients.
A is known in linear algebra as Sylvester type matrix.
Example:
Write the identity equation for a system of 3nn ba and 1n t .
Solution:
We have F.A+B.G=T
21nn bf
21nn ag
)ZaZ.aZa1).(Z.fZf1(
3
.3
2
2
1
.1
2
2
1
.1
1
1
2
2.
1
1.o
3
.3
2
2
1
.1 Z.t1)Z.gZ.gg).(ZbZ.bZb(
Sub. in equation (15), to get:
33
2323
12312
121
1
b00a0
bb0aa
bbbaa
0bb1a
00b01
.
2
1
o
2
1
g
g
g
f
f
=
0
0
a
a
at
3
2
11
A c b
The above equation can be solved in the normal way by inverting A to obtain
the vector of controller parameters:
c=A-1
.b (16)
Equation (13) substituted in equation (12) gives the closed loop equation with
desired poles given by T, thus:
)z(rT
H.B)z(y (17)
and this gives the desired stability characteristic.
Adaptive Control
30
However, we still have the problem of ensuring that the output y(z) is equal to
the reference r(z) for constant (or slowly changing) reference signal. The
simplest way is to select:
|B
TH
1Z
(18)
let hH (h is a constant value)
This choice is made so that the closed loop transfer function )T
h.B( will be (1) at
zero frequency and y(z)=r(z) for constant reference signal as required.
An Alternative procedure:
An Alternative procedure is to choose H to give the desired system zeros.
The point of this is that the zeros of a system, which in this case are the roots of
(B.H=0), in fact also influence the transient response shape. The immediate
answer is to cancel the B term by making:
B
1H (19)
But, the B part of a system can be inverse unstable, (the roots of B lie outside
the unit circle), so cancellation must be exact, since any slight error in
cancellation will leave an unstable component in the relationship between r(z)
and y(z).
Time delays:
Any time delay leads to some leading zeros coefficients in B plus possible
non-minimum phase zeros associated with the partial time delay, m. If there
are know to be (k) zero coefficients then it is useful to use the information and
substitute B.ZK for B in the relevant relations. The pole-assignment identity
will now take the form:
TG.B.ZF.AK (20)
Adaptive Control
31
This equation is solved as before except that the K top rows of the B block will
be zero because of the KZ
terms.
Three-term Controller design by pole-assignment:
Three-term or PID controller combines proportional action, integral
action and derivative action and can put into the form:
)z(eZ1
Z.gZ.gg)z(u
1
2
2
1
1o
(21)
The coefficients go, g1 and g2 are related to the proportional derivative and
integral gain setting, (assuming backward shift approximation) with Ts=1
second):
Kp=- g1 – 2. g2
Kd=g2
Ki=go + g1 + g2 (H.W. Check ?)
In order to synthesize exactly the PID controller coefficients, we can assume
the system to be controlled has the following structure:
)z(u.Z.aZ.a1
Z.b)z(y
2
2
1
1
1
1
(22)
and we can assume that a1, a2 and b1 can be estimated by RLS method.
The restriction on the system model form is to ensure that only one set of PID
controller coefficients arises from the design. Combine equations (21) and (22),
we get:
1
1
2
2
1
1o
2
2
1
1
1
2
2
1
1o
1
.1
Z.b).Z.gZ.gg()Z.aZ.a1).(Z1(
)z(r).Z.gZ.gg.(Zb)z(y
(23)
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We can now select the coefficients go, g1 and g2 to give a desired closed loop
performance. Suppose we have selected a desired closed loop the rise time and
damped natural frequency and hence arrived at the corresponding desired
closed loop T polynomial:
2
2
1
1 Z.tZ.t1T (24)
Thus:
2
2
1
1
1
1
2
2
1
1o
2
2
1
1
1Z.tZ.t1Z.b).Z.gZ.gg()Z.aZ.a1).(Z1(
(25)
By equating coefficients of like power of Z, we get:
1
11
ob
)a1(tg
,
1
212
1b
)aa(tg
and
1
2
2b
ag (26)
The steady state matching of y(z) to r(z) for a constant reference signal is
ensured by the factor )Z1(1 in the denominator of equation (23).
Specifically, for zero frequency (Z=1), so that independent of the system or
controller parameters, we have y(z)=r(z) (for constant r(z))
Self-Tuning pole assignment algorithm:
The block diagram below shows the layout of a pole assignment self
tuner in which a recursive estimator is combined with a pole assignment
synthesis rule in order to continuously update the controller coefficients.
Certain performance requirements have fed into the synthesis block. These
consist of the following controller design information:
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Fig. Self-Tuning Pole Assignment System.
a) The desired closed loop pole set, specified by T.
b) The form of the controller e.g. whether it is a servo system, a regulator
or a combination of both.
In addition, further information must be supplied concerning the
configuration requirements of the self-tuner, including:
c) The sampler rate to be used.
d) The degree of the system model polynomials na, nb ….etc.
e) The delay K (for delayed systems) in the system, if known.
Provided with these information, the self –tuning system can be setup to go
through the following cycle of adaptation:
Self-Tuning cycle:
At each sample interval Ts, the following sequence of action is taken:
Step (i) Data capture:
The system output y(KT), reference input r(KT) and any other variables
of importance are measured.
Step (ii) Estimator update:
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The data acquired in (i) is used together with past data and the previous
control signal to update the parameter estimates in a model of the system
using an appropriate recursive estimator.
Step (iii) Controller synthesis:
The updated parameters from (ii) are used in a pole assignment identity
to synthesize the parameters of the desired controller.
Step (iv) Control algorithm:
The controller parameters synthesized in (iii) are used in a controller to
calculate and input the next control signal u(KT).
At the end of the cycle the control computer waits until the end of sample
interval T and then repeats the cycle for interval T+1, and so on.
Steps in the self-tuning cycle have computed sequentially. However,
figure below illustrates this in terms of a timing and sequence diagram. The
total computation time must be less than the sample interval and is
generally assumed to be much less.
Fig. Timing and sequence diagram for self-tuning controller.
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Hint: The conversion from s-plane into z-plane can be done using digital
approximations. Use the following conversions:
Continuous function sdt
dy
Forward difference: ST
1zs
Backward difference: z.T
1zs
S
Central difference: z.T.2
1zs
S
2
Where: sT is the sampling time in seconds.
Tutorials:
Question (1):
Derive the PID controller equation in z-domain using backward
approximation.
Question (2):
Calculate the desired closed loop equation in discrete form for a second order
system with 10n , 7.0 and the sampling time is 0.1 second.
Question (3):
For the following model:
)z(uZ72.0Z7.11
Z2Z)z(y
21
21
Design a digital controller using pole placement that satisfies the following
C\Cs equation: 1Z904.01T
.
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c- Model-Reference Adaptive Controller (MRAC):
The mode-Reference Adaptive Controller is one of the main approaches of
adaptive control. The basic principle is illustrated in figure below:
Fig. Block diagram of RMAC.
The desired performance is expressed in terms of a reference model, which
gives the desired response to a command signal. The system also has an
ordinary feedback loop composed of the process and the regulator. The error
(e) is the difference between the output(s) of the system and the reference
model. The regulator has parameters that are changed, based on the error.
There are thus two loops: an inner loop, which provides the ordinary control
feedback and outer loop, which adjusts the parameters in the inner loop. The
inner loop is assumed faster than the outer loop.
In this adaptive system, two ideas were introduced. First, the performance of
the system is specified by a model. Second, the parameters of the regulator are
adjusted based on the error between the reference model and the system.
There are essentially three basic approaches to analysis and design of MRAC:
1- The gradient approach.
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2- Lyapunov functions.
3- Passivity theory.
For a system with adjustable parameters, the model reference adaptive
methods gives a general approach for adjusting the parameters so that the
closed loop transfer function will be close to a prescribed model. This is called
the Model-following problem.
One important question is how small we can make the error (e)?. This depends
both on the model, the system and the command signal. If it is possible to make
the error equal to zero for all command signals, the perfect model following is
achieved.
Model Following:
The model-Following problem can be solved using pole placement design.
Model –Following is a simple and useful way to formulate and solve a servo
control problem. The basic idea is very simple. Servo performance is specified
indirectly by giving a mathematical model for the desired response. The
specified model can be linear as well as nonlinear. The parameters in the
system are adjusted in order to get the output of the process model (y) as close
as possible to the reference model output (ym) for a given class of input signals.
Optimization methods are thus natural tool in MRAC design.
The Gradient approach:
This is a fundamental idea in the MRAC approach. The parameters
adjustment scheme is usually called MIT rule.
The MIT rule:
Assume that we attempt to change the parameters of the regulator so that the
error between the output(s) of the process and the reference model is driven to
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zero. Let (e) denote the error and () represents the parameters. Introduce the
criterion:
2e
2
1)(J (27)
where J is called the loss function. To make J small, it is possible to change the
parameters in the direction of the negative gradient of (J). i. e.:
(28)
It is assumed that the parameters change much more slowly than the other
variables in the system, then the derivative
ecan be evaluated under the
assumption that is constant. The derivative
eis the sensitivity derivative of
the system. The adjustment rule of equation (28), where
eis the sensitivity
derivative, is commonly referred to as the MIT rule. The choice of loss function
is, of course, arbitrary. If the loss function is chosen as:
e)(J
Thus, the adjustment rule becomes:
)e(signe
dt
d
Equation (28) also applies to the case of many adjustable parameters. The
variable should be interpreted as a vector and
eis as the gradient of the
error with respect to the parameters.
n321i
......
n321i
e......
eeee
Je
.e.J
dt
d
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for 2
e2
1)(J :
n321
e......
eee.e.
dt
d
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Note:
In the MIT rule, it is not necessary to require perfect model-following. The
procedure can be applied to nonlinear systems. The method can also be used to
handle partially known systems.
The MIT rule will perform well if the adaptation gain () is small.
Consequently, it is not possible to give fixed limits that guarantee stability. The
MIT rule can thus give an unstable closed loop system.
Lyapunov’s Second method (1892):
Lyapunov introduced an interesting direct method to investigate the stability
of a solution to a nonlinear differential equation. The key idea is illustrated in
figure below:
Fig. Illustration of Lyaunov’s method for investigating stability.
The equilibrium will be stable if we can find a real function on the state space
whose level curves enclose the equilibrium such that the derivative of the state
variable always towards the interior of the level curves.
To state the results formally, let the differential equation be:
)t,x(fx.
and
f(0,t)=0
where x is a state vector of dimension n. It is assumed that f is such that
solutions exist for all o
tt . The equilibrium point o
t at the origin. This
involves no loss of generality since this can be achieved through a simple
coordinate transformation.
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Lyapunov Stability Theorem:
Let the function RR:V1n
satisfy the conditions:
1. 0)t,0(V for all t R.
2. V is differentiable in X and t.
3. V is positive definite.
4. V.
is a negative definite (or negative Semi-definite).
Then the equilibrium state at the origin is uniformly asymptotically stable.
If )t,x(V as ,x then equilibrium state at the origin is uniformly
asymptotically stable in the large.
Then V is called Lyapunov function.
Note:
a. Positive definiteness of a scalar function: A scalar function V(x) is said to be
positive definite in a region includes the origin if v(x) > 0 for all nonzero
states x and v(0)=0.
b. Negative definiteness scalar function: A scalar function V(x) is said to be
negative definite if –V(x) is positive definite.
c. Positive semi-definiteness of a scalar function: A scalar function V(x) is said
to be positive semi-definite if it is positive at all states except at the origin and
at certain other states where it is zero.
d. Negative Semi-definiteness of a scalar function: A scalar function V(x) is
said to be negative semi-definite if –V(X) is positive semi-definite.
Example:
For the following matrix (P):
Pnn....1Pn
......
......
n2P...22P21P
n1P...12P11P
P
if P11 > 0, 22P21P
12P11P>0, . . . . . . . . . . . . . . . . . . . . .
Pnn.....
......
......
.....21P
n1P...12P11P
> 0
Then P is positive definite.
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Expert Control:
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Neural Networks:
In an expert system, the control law is a logic function that gives the
control action as a function of senor patterns. The function is adaptive in the
sense that it will adjust itself automatically.
The Perceptorn proposed by Rosenbatt (1959) is one way to obtain a learning
function. To describe the perceptorn , let (ui with i=1,2,3,…., n) be inputs and
(yi with i=1,2,3,…, n) be outputs. In the perceptron, the output is formed as:
n
1jjiji
)bu.w(fy , i=1,2,3,…,n
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where wij is the weight, b is the bias and f is the threshold function.
As an example:
0ifx0
0ifx1)x(f
To update the weight, the perceptron uses a very simple idea, which is called
Hebb’s principle. Apply a given pattern to the inputs and clamp the outputs to
the desired response, then increase the weights between nodes that are
simultaneously excited. This principle was formulated in Hebb (1949), in an
attempt to model neuron networks:
)yy).(t(u.)t(wwj
0
jiijij
where 0
jy is the desired response and
jy is the response predicated by the
model. By regarding the weight as a parameter, it becomes clear that updating
formula is identical to gradient method for parameter estimation.
There are several ways to use Neural Network in control systems. In the following,
some of these methods:
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